. The ﬂuid
velocity v at x is deﬁned by the average value of the molecular veloc-
ities, in such a way
v(x) = u
α
:=
¸
α
m
α
u
α
¸
α
m
α
, (1.2)
where m
α
= m (by the assumption),
¸
α
m
α
= mN
∆
= ρ∆V , and
· denotes an average with respect to the molecules concerned. The
diﬀerence ˜ u
α
= u
α
− v is called the peculiar velocity or thermal
velocity.
In the kinetic theory of molecules, the temperature T is deﬁned
by the law that the average of peculiar kinetic energy per degree-of-
freedom is equal to
1
2
kT, where k is the Boltzmann constant.
2
Each
molecule has three degrees of freedom for translational motion. It is
assumed that

is given by kT from (1.3). Thus, from (1.4) and (1.6), we obtain
p(x) = NkT. (1.7)
This is known as the equation of state of an ideal gas.
4
The density ρ(x), velocity v(x), temperature T(x) and pressure
p(x) thus deﬁned depend on the position x = (x, y, z) and the time t
smoothly, since the molecular kinetic motion usually works to smooth
out discontinuity (if any) by the transport phenomena considered
in Chapter 2. Namely, these variables are regarded as continuous
and in addition diﬀerentiable functions of (x, y, z, t). Such variables
are called ﬁelds. This point of view is often called the continuum
hypothesis.
From a mathematical aspect, ﬂow of a ﬂuid is regarded as a con-
tinuous sequence of mappings. Consider all the ﬂuid particles compos-
ing a subdomain B
0
at an initial instant t = 0. After an inﬁnitesimal
time δt, a particle at x ∈ B moves from x to x +δx:
x → x +δx = x +vδt +O(δt
2
) (1.8)
by the ﬂow ﬁeld v(x, t). Then the domain B
0
may be mapped to
B
δt
(say). Subsequent mapping occurs for another δt from B
δt
to
B
2δt
, and so on. In this way, the initial domain B
0
is mapped one
after another smoothly and constantly. At a later time t, the domain
4
For a gram-molecule of an ideal gas, N is replaced by N
A
= 6.023 × 10
23
(Avogadro’s constant). The product N
A
K = R is called the gas constant:
R = 8.314 × 10
7
erg/deg. For an ideal gas of molecular weight µ
m
, the equa-
tion (1.7) reduces to p = (1/µ
m
)ρRT, where ρ = mN, µ
m
= mN
A
and R = N
A
k.
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6 Flows
B
0
would be mapped to B
t
. The map might be diﬀerentiable with
respect to x, and in addition, for such a map, there is an inverse
map. This kind of map is termed a diﬀeomorphism (i.e. diﬀerentiable
homeomorphism).
1.3. Stream-line, particle-path and streak-line
1.3.1. Stream-line
Suppose that a velocity ﬁeld v(x, t) = (u, v, w) is given in a sub-
domain of three-dimensional Euclidean space R
3
, and that, at a
given time t, the vector ﬁeld v = (u, v, w) is continuous and smooth
at every point (x, y, z) in the domain. It is known in the the-
ory of ordinary diﬀerential equations in mathematics that one can
draw curves so that the curves are tangent to the vectors at all
points. Provided that the curve is represented as (x(s), y(s), z(s))
in terms of a parameter s, the tangent to the curve is written as
(dx/ds, dy/ds, dz/ds), which should be parallel to the given vector
ﬁeld (u(x, y, z), v(x, y, z), w(x, y, z)) by the above deﬁnition. This is
written in the following way:
dx
u(x, y, z)
=
dy
v(x, y, z)
=
dz
w(x, y, z)
= ds. (1.9)
This system of ordinary diﬀerential equations can be integrated for a
given initial condition at s = 0, at least locally in the neighborhood
of s = 0. Namely, a curve through the point P = (x(0), y(0), z(0)) is
determined uniquely.
5
The curve thus obtained is called a stream-line.
For a set of initial conditions, a family of curves is obtained. Thus,
a family of stream-lines are deﬁned at each instant t (Fig. 1.1).
5
Mathematically, existence of solutions to Eq. (1.9) is assured by the continuity
(and boundedness) of the three component functions of v(x, t). For the unique-
ness of the solution to the initial condition, one of the simplest conditions is the
Lipschitz condition: |v(x, t) −v(y, t)| K|x −y| for a positive constant K.
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1.3. Stream-line, particle-path and streak-line 7
Fig. 1.1. Stream-lines.
1.3.2. Particle-path (path-line)
Next, let us take a particle-wise point of view. Choosing a ﬂuid par-
ticle A, whose position was at a = (a, b, c) at the time t = 0, we
follow its subsequent motion governed by the velocity ﬁeld v(x, t) =
(u, v, w). Writing its position as X
a
(t) = (X(t), Y (t), Z(t)), equations
of motion of the particle are
dX
dt
= u(X, Y, Z, t),
dY
dt
= v(X, Y, Z, t),
dZ
dt
= w(X, Y, Z, t).
(1.10)
This can be solved at least locally in time, and the solution would
be represented as X
a
(t) = X(a, t) = (X(t), Y (t), Z(t)), where
X(t) = X(a, b, c, t), Y (t) = Y (a, b, c, t),
Z(t) = Z(a, b, c, t),
(1.11)
and X
a
(0) = a. For a ﬁxed particle speciﬁed with a = (a, b, c), the
function X
a
(t) represents a curve parametrized with t, called the
particle path, or a path-line. Correspondingly, the particle velocity is
given by
V
a
(t) =
d
dt
X
a
(t) =
∂
∂t
X(a, t) = v(X
a
, t). (1.12)
This particle-wise description is often called the Lagrangian descrip-
tion, whereas the ﬁeld description such as v(x, t) for a point x and
a time t is called the Eulerian description.
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8 Flows
It is seen that the two equations (1.9) and (1.10) are identical
except the fact that the right-hand sides of (1.10) include the time t.
Hence if the velocity ﬁeld is steady, i.e. v does not depend on t, then
both equations are equivalent, implying that both stream-lines and
particle-paths are identical in steady ﬂows.
1.3.3. Streak-line
In most visualizations of ﬂows or experiments, a common practice
is to introduce dye or smoke at ﬁxed positions in a ﬂuid ﬂow and
observe colored patterns formed in the ﬂow ﬁeld (Fig. 1.2). Smoke
from a chimney is another example of analogous pattern. An instan-
taneous curve composed of all ﬂuid elements that have passed the
same particular ﬁxed point P at previous times is called the streak-
line.
Denoting the ﬁxed point P by A, the ﬂuid particle that has passed
the point A at a previous time τ will be located at X = X(a
τ
, t)
at a later time t where a
τ
is deﬁned by A = X(a
τ
, τ). Thus the
streak-line at a time t is represented parametrically by the function
X(A, t −τ) with the parameter τ.
If the ﬂow ﬁeld is steady (Fig. 1.3), it is obvious that the streak-
line coincides with the particle-path, and therefore with the stream-
line. However, if the ﬂow ﬁeld is time-dependent (Fig. 1.4), then all
the three lines are diﬀerent, and they appear quite diﬀerently.
1.3.4. Lagrange derivative
Suppose that the temperature ﬁeld is expressed by T(x, t) and
that the velocity ﬁeld is given by v(x, t), in the way of Eulerian
description. Consider a ﬂuid particle denoted by the parameter a in
the ﬂow ﬁeld and examine how its temperature T
a
(t) changes during
the motion. Let the particle position be given by X
a
(t) = (X, Y, Z)
and its velocity by v
a
(t) = (u
a
, v
a
, w
a
). Then the particle tempera-
ture is expressed by
T
a
(t) = T(X
a
, t) = T(X(t), Y (t), Z(t); t).
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1.3. Stream-line, particle-path and streak-line 9
Fig. 1.2. Visualization of the wake behind a thin circular cyinder (of diame-
ter 5 mm) by a smoke-wire method. The wake is the central horizontal layer of
irregular smoke pattern, and the many parallel horizontal lines in the upper and
lower layers show a uniform stream of wind velocity 1 m/s from left to right. The
smoke lines originate from equally-spaced discrete points on a vertical straight
wire on the left placed at just upstream position of the cylinder (at the point of
intersection of the central horizontal white line (from the left) and the vertical
line connecting the two arrows out of the frame). Thus, all the smoke lines are
streak-lines. The illumination is from upward right, and hence the shadow line of
the cylinder is visible to downward left on the lower left side. The regular peri-
odic pattern observed in the initial development of the wake is the K´arm´an vortex
street. The vertical white line at the central right shows the distance 1 m from
the cylinder. This is placed in order to show how the wake reorganizes to another
periodic structure of larger eddies. [As for the wake, see Problem 4.6 (Fig. 4.12),
and Fig. 4.7.] The photograph is provided through the courtesy of Prof. S. Taneda
(Kyushu University, Japan, 1988). R
e
= 350 (see Table 4.2).
Hence, the time derivative of the particle temperature is given by
d
dt
T
a
(t) =
∂T
∂t
+
dX
dt
∂T
∂x
+
dY
dt
∂T
∂y
+
dZ
dt
∂T
∂z
=
∂T
∂t
+u
a
∂T
∂x
+v
a
∂T
∂y
+w
a
∂T
∂z
=

∂
∂t
+u
∂
∂x
+v
∂
∂y
+w
∂
∂z

T

x=X
a
.
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10 Flows
stream-line
p
a
rtic
le
-p
a
th
streak-line
dye
S
•
Fig. 1.3. Steady ﬂow: stream-lines (thin solid lines), particle-path (broken lines),
and streak-lines (a thick solid line).
P
The tangents of the
stream-line and
streak-line
coincide.
The tangents of the
stream-line and
particle-path
coincide.
streak-line
stream-line
stream-line at t
p
a
r
t
i
c
l
e
-
p
a
t
h
Fig. 1.4. Unsteady ﬂow: stream-lines (thin solid lines), particle-path (a bro-
ken line), and streak-line (a thick solid line). The particle P started from the
ﬁxed point O at a time t
0
and is now located at P at t after the times t
1
, t
2
and t
3
.
It is convenient to deﬁne the diﬀerentiation on the right-hand side
by using the operator,
D
Dt
:=
∂
∂t
+u
∂
∂x
+v
∂
∂y
+w
∂
∂z
= ∂
t
+u∂
x
+v∂
y
+w∂
z
,
which is called the convective derivative, where ∂
t
:= ∂/∂t, ∂
x
:=
∂/∂x, and so on. As is evident from the above derivation, this time
derivative denotes the diﬀerentiation following the particle motion.
This derivative is called variously as the material derivative, con-
vective derivative or Lagrange derivative. Thus, we have dT
a
/dt =
DT/Dt

, (1.16)
where ∂
k
v
i
= ∂v
i
/∂x
k
. This can be also written as
6
δv
i
=
3
¸
k=1
s
k
∂
k
v
i
= s
k
∂
k
v
i
. (1.17)
The term ∂
k
v
i
can be decomposed into a symmetric part e
ik
and
an anti-symmetric part g
ik
in general (Stokes (1845), [Dar05]),
deﬁned by
e
ik
=
1
2
(∂
k
v
i
+∂
i
v
k
) = e
ki
, (1.18)
g
ik
=
1
2
(∂
k
v
i
−∂
i
v
k
) = −g
ki
. (1.19)
Then one can write as ∂
k
v
i
= e
ik
+g
ik
. Using e
ik
and g
ik
, the velocity
diﬀerence δv
i
is decomposed as δv
i
= δv
(s)
i
+δv
(a)
i
, where
δv
(s)
i
= e
ik
s
k
, (1.20)
δv
(a)
i
= g
ik
s
k
. (1.21)
These components represent two fundamental modes of relative
motion, which we will consider in detail below.
6
The summation convention is assumed here, which takes a sum with respect to
the repeated indices such as k. Henceforth, summation is meant for such indices
without the symbol
P
3
k=1
.
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1.4. Relative motion 13
1.4.2. Symmetric part (pure straining motion)
The symmetric part is written as

¸
¸
¸
¸
δv
(s)
1
δv
(s)
2
δv
(s)
3
¸

=

¸
e
11
e
12
e
13
e
12
e
22
e
23
e
13
e
23
e
33
¸

¸
s
1
s
2
s
3
¸

. (1.22)
Any symmetric (real) matrix can be made diagonal by a coordi-
nate transformation (called the orthogonal transformation, see the
footnote 7) to a principal coordinate frame. Using capital letters to
denote corresponding variables in the principal frame, the expression
(1.22) is transformed to

s
j
.
In the vector notation, using (A.12), this is written as
δv
(a)
=
1
2
ω ×s. (1.30)
8
For the deﬁnition of ε
ijk
, see Appendix A.1. For example, we have g
12
=
−
1
2
(ε
121
ω
1
+ ε
122
ω
2
+ ε
123
ω
3
) = −
1
2
ω
3
.
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1.5. Problems 15
This component of relative velocity describes a rotation of the angu-
lar velocity
1
2
ω. Although ω depends on x, it is independent of the
displacement vector s. Namely, every point s in the neighborhood of
x rotates with the same angular velocity. Thus, it is found that δv
(a)
represents local rigid-body rotation.
In summary, it is found that the local relative velocity δv con-
sists of a pure straining motion δv
(s)
and a local rigid-body rotation
δv
(a)
.
1.5. Problems
Problem 1.1 Pattern of ink-drift
Suppose that some amount of water is contained in a vessel, and
the water is set in motion and its horizontal surface is in smooth
motion. Let a liquid-drop of Chinese ink be placed quietly on the
ﬂat horizontal surface maintaining a ﬂow with some eddies. The ink
covers a certain compact area of the surface.
After a while, some ink pattern will be observed. If a sheet of
plain paper (for calligraphy) is placed quietly on the free surface of
the water, a pattern will be printed on the paper, which is called
the ink-drift printing (Fig. 1.5). This pattern is a snap-shot at an
instant and consists of a number of curves. What sort of lines are the
curves printed on the paper? Are they stream-lines, particle-paths or
streak-lines, or other kind of lines?
Problem 1.2 Divergence operator div
Consider a small volume of ﬂuid of a rectangular parallelepiped in a
ﬂow ﬁeld of ﬂuid velocity v = (v
x
, v
y
, v
z
). The ﬂuid volume V changes
under the straining motion. Show that the time-rate of change of
volume V per unit volume is given by the following,
1
V
dV
dt
= div v =
∂v
x
∂x
+
∂v
y
∂y
+
∂v
z
∂z
. (1.31)
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16 Flows
Fig. 1.5. Ink-drift printing.
Problem 1.3 Acceleration of a ﬂuid particle
Given the velocity ﬁeld v(x, t) with x = (x, y, z) and v = (u, v, w).
Show that the velocity and acceleration of a ﬂuid particle are given
by the following expressions:
D
Dt
x = v, (Sec. 12.6.2) (1.32)
D
Dt
v = ∂
t
v + (v · ∇)v. (1.33)
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Chapter 2
Fluids
2.1. Continuum and transport phenomena
The motion of a ﬂuid is studied on the basis of the fundamental prin-
ciple of mechanics, namely the conservation laws of mass, momentum
and energy. For a state of matter to which the continuum hypothesis
(Sec. 1.2) can be applied, macroscopic motions of the matter (a ﬂuid)
are less sensitive to whether the structure of matter is discrete or
continuous. In the continuum representation of ﬂuids, the eﬀect of
actual discrete molecular motion is taken into account as transport
phenomena such as diﬀusion, viscosity and thermal conductivity in
equations of motion. In ﬂuid mechanics, all variables, such as mass
density, momentum, energy and thermodynamic variables (pressure,
temperature, entropy, enthalpy, or internal energy, etc.), are regarded
as continuous and diﬀerentiable functions of position and time.
Equilibrium in a material is represented by the property that the
thermodynamic state-variables take uniform values at all points of
the material. In this situation, each part of the material is assumed
to be in equilibrium mechanically and thermodynamically with the
surrounding medium. However, in most circumstances where real ﬂu-
ids are exposed, the ﬂuids are hardly in equilibrium, but state vari-
ables vary from point to point. When the state variables are not
uniform, there occurs exchange of physical quantities dynamically
and thermodynamically. Usually when external forcing is absent, the
matter is brought to an equilibrium in most circumstances by the
exchange. This is considered to be due to the molecular structure or
17
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18 Fluids
due to random interacting motion of uncountably many molecules.
The entropy law is a typical one in this regard.
For conservative quantities, the exchange of variables make per-
fect sense. Because, with conservative variables, it is possible to con-
nect the decrease of some quantity at a point to the increase of the
same quantity at another point, and the exchange is understood as
the transfer. This type of exchange is called the transfer phenomenon,
or transport phenomenon. Therefore, corresponding to the three con-
servation laws mentioned above, we have three transfer phenomena:
mass diﬀusion, momentum diﬀusion and thermal diﬀusion.
2.2. Mass diﬀusion in a ﬂuid mixture
Diﬀusion in a ﬂuid mixture occurs when composition varies with posi-
tion. Suppose that the concentration of one constituent β of matter
is denoted by C which is the mass proportion of the component with
respect to the total mass ρ in a unit volume. Hence, the mass den-
sity of the component is given by ρC, and C is assumed to be a
diﬀerentiable function of point x and time t: C(x, t).
Within the mixture, we choose an arbitrary surface element δA
with its unit normal n. Diﬀusion of the component β through the
surface δA occurs from one side to the other and vice versa. However,
owing to the nonuniformity of the distribution C(x), there is a net
transfer as a balance of the two counter ﬂuxes (Fig. 2.1). Let us
write the net transfer through δA(n) toward the direction n per unit
time as
q(x) · nδA(n), (2.1)
C (x
1
)
C (x
2
)
n
Fig. 2.1. Two counter ﬂuxes.
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2.2. Mass diﬀusion in a ﬂuid mixture 19
assuming that it is proportional to the area δA, and a vector q can
be deﬁned at each point x, where q is called the diﬀusion ﬂux.
From the above consideration, the diﬀusion ﬂux q will be related
to the concentration C. Since the ﬂux q should be zero if the concen-
tration is uniform, q would depend on the concentration gradient or
derivatives of C. Provided that the concentration gradient is small,
the ﬂux q would depend on the gradient linearly with the propor-
tional constant k
C
as follows:
q
(C)
= −k
C
grad C = −k
C
(∂
x
C, ∂
y
C, ∂
z
C), (2.2)
where k
C
is the coeﬃcient of mass diﬀusion. This is regarded as
a mathematical assumption that higher-order terms are negligible
when the ﬂux is represented by a Taylor series with respect to deriva-
tives of C. From the aspect of molecular motion, a macroscopic scale
is much larger than the microscopic intermolecular distance, so that
the concentration gradient in usual macroscopic problems would be
very small from the view point of the molecular structure. The above
expression (2.2) is valid in an isotropic material. In an anisotropic
medium, the coeﬃcient should be a tensor k
ij
, rather than a scalar
constant k
C
. The coeﬃcient k
C
is positive usually, and the diﬀusion
ﬂux is directed from the points of larger C to those of smaller C,
resulting in attenuation of the degree of C nonuniformity.
Next, in order to derive an equation governing C, we choose an
arbitrary volume V bounded by a closed surface A in a ﬂuid mixture
at rest (Fig. 2.2), and observe the volume V with respect to the frame
of the center of mass. So that, there is no macroscopic motion. The
total mass of the component β in the volume V is M
β
=

V
ρC dV
by the deﬁnition of C. Some of this component will move out of V
A
n
V
Fig. 2.2. An arbitrary volume V .
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20 Fluids
by the diﬀusion ﬂux q through the bounding surface A, the total
amount of outward diﬀusion is given by

A
q
(C)
· ndA = −

A
k
C
n · grad C dA,
where n is unit outward normal to the surface element dA. This
outward ﬂux gives the rate of decrease of the mass M
β
(per unit
time). Hence we have the equation for the rate of increase of M
β
:
d
dt

V
ρC dV =

V
∂
∂t

ρC

dV =

A
k
C
n · ∇C dA,
where the time derivative is placed within the integral sign since
the volume element dV is ﬁxed in space, and grad is replaced by ∇.
Applying the Gauss’s theorem (see Sec. 3.1 and Appendix A.6) trans-
forming the surface integral into a volume integral, we obtain

V
¸
∂
∂t
(ρC) −div(k
C
∇C)

dV = 0.
Since this relation is valid for any volume V , the integrand must
vanish identically. Thus we obtain
∂
∂t
(ρC) = div(k
C
∇C). (2.3)
In a ﬂuid at rest in equilibrium, no net translation of mass is pos-
sible. Therefore, the total mass ρ in unit volume is kept constant.
1
Moreover, the diﬀusion coeﬃcient k
C
is assumed to be constant. In
this case, the above equation reduces to
∂C
∂t
= λ
C
∆C, λ
C
=
k
C
ρ
, (2.4)
where ∆ is the Laplacian operator,
∆ := ∇
2
=
∂
2
∂x
2
+
∂
2
∂y
2
+
∂
2
∂z
2
.
1
When the diﬀusing component is only a small fraction of total mass, the density ρ
may be regarded as constant even when the frame is not of the center of mass.
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2.3. Thermal diﬀusion 21
Equation (2.4) is the diﬀusion equation, and λ
C
is the diﬀusion
coeﬃcient.
2.3. Thermal diﬀusion
Transport of the molecular random kinetic energy (i.e. the heat
energy) is called heat transfer. A molecule in a gas carries its own
kinetic energy. The average kinetic energy of molecular random
velocities is the thermal energy, which deﬁnes the temperature T
(Sec. 1.2).
Choosing an imaginary surface element δA in a gas, we consider
such molecules moving from one side to the other, and those vice
versa. If the temperatures on both sides are equal, then the transfer
of thermal energy (from one side to the other) cancels out with the
counter transfer, and there is no net heat transfer. However, if the
temperature T depends on position x, obviously there is a net heat
transfer. The ﬂow of heat through the surface element δA will be
written in the form (2.1), where the vector q is now called the heat
ﬂux. In liquids or solids, heat transport is caused by collision or
interaction between neighboring molecules.
Analogously with the mass diﬀusion, the heat ﬂux will be repre-
sented in terms of the temperature gradient as
q = −k grad T, (2.5)
where k is termed the thermal conductivity. The second law of ther-
modynamics (for the entropy) implies that the coeﬃcient k should
be positive (see Sec. 4.2).
The equation corresponding to (2.3) is written as
ρC
p
∂T
∂t
= div(k∇T),
since the increase of heat energy is given by ρC
p
∆T for a temperature
increase ∆T, where C
p
is the speciﬁc heat per unit mass at constant
pressure.
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22 Fluids
Corresponding to (2.4), the equation of thermal conduction is
given by
∂T
∂t
= λ
T
∆T, λ
T
=
k
ρc
p
, (2.6)
in a ﬂuid at rest, where λ
T
is the thermal diﬀusivity. Equation (2.6)
is also called the Fourier’s equation of thermal conduction.
2.4. Momentum transfer
Transfer of molecular momentum emerges as an internal friction. A
ﬂuid with such an internal friction is said to be viscous. The momen-
tum transfer is caused by molecules carrying their momenta, or by
interacting force between molecules. The concentration and temper-
ature considered above were scalars, however momentum is a vector.
This requires some modiﬁcation in the formulation of momentum
transfer.
Macroscopic velocity v of a ﬂuid at a point x in space is deﬁned
by the velocity of the center of mass of a ﬂuid particle located at x
instantaneously. Constituent molecules in the ﬂuid particle are mov-
ing randomly with velocities ˜ u
α
(Sec. 1.2). Let us consider the trans-
port of the ith component of momentum. Instead of the expression
(2.1), the ith momentum transfer through a surface element δA(n)
from the side I (to which the normal n is directed) to the other II is
deﬁned (Fig. 2.3) as
q
ij
n
j
δA(n), (2.7)
Fig. 2.3. Momentum transfer through a surface element δA(n).
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2.4. Momentum transfer 23
where q
ij
n
j
=
¸
3
j=1
q
ij
n
j
. The tensor quantity q
ij
represents the
ith component of momentum passing per unit time through a unit
area normal to the jth axis. The dimension of q
ij
is equivalent to
that of force per unit area, and such a quantity is termed a stress
tensor. The stress associated with nonuniform velocity ﬁeld v(x) is
characterized by a tangential force-component to the surface element
considered, and called the viscous stress. It can be veriﬁed that the
stress tensor must be symmetric (Problem 2.3):
q
ij
= q
ji
.
Concerning the momentum transfer, there is another signiﬁcant
diﬀerence from the previous cases of the transfer of concentration
or temperature. Suppose that the ﬂuid is at rest and is in both
mechanical and thermodynamical equilibrium. Hence, variables are
distributed uniformly in space. Let us pay attention to a neighbor-
hood on one side of a surface element δA(n) where the normal vector
n is directed. In the case of heat, the heat ﬂux escaping out of δA is
balanced with the ﬂux coming in through δA, resulting in vanishing
net ﬂux in the equilibrium. How about in the case of momentum?
The negative momentum (because it is anti-parallel to n) escaping
from from the side I out of δA would be expressed as “vanishing of
negative momentum” Q, while the positive momentum coming into
the side I through δA would be expressed as “emerging of positive
momentum” P which is a contraposition of the previous statement.
Hence, both are same and we have twice the positive momentum
gain P (stress). However, on the other side of the surface δA, the
situation is reversed and we have twice the loss of P. Thus, both
stresses counter balance. This is recognized as the pressure.
In a ﬂuid at rest, the momentum transfer is given by
q
ij
= −pδ
ij
, hence q
ij
n
j
= −pn
i
, (2.8)
(see Eq. (4.1)), where δ
ij
is the Kronecker’s delta and the minus
sign is due to the deﬁnition of q
ij
(see the footnote to Sec. 4.1 and
Appendix A.1 for δ
ij
). In a uniform ﬂuid, the pressure is always
normal to the surface chosen (Fig. 2.4). Total pressure force acting
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24 Fluids
Fig. 2.4. Pressure stress.
on a ﬂuid particle is given by
−

S
p
pn
i
dA, (2.9)
where S
p
denotes the surface of a small ﬂuid particle.
In the transport phenomena considered above such as diﬀusion of
mass, heat or momentum, the net transfers are in the direction of
diminishing nonuniformity (Sec. 4.2). The coeﬃcients of diﬀusivity,
thermal conductivity and viscosity in the representation of ﬂuxes are
called the transport coeﬃcients.
2.5. An ideal ﬂuid and Newtonian viscous ﬂuid
The ﬂow of a viscous ﬂuid along a smooth solid wall at rest is char-
acterized by the property that the velocity vanishes at the wall sur-
face. If the velocity far from the wall is large enough, the proﬁle of
the tangential velocity distribution perpendicular to the surface has
a characteristic form of a thin layer, termed as a boundary layer.
Suppose there is a parallel ﬂow along a ﬂat plate with the velocity
far from it being U in the x direction, the y axis being taken perpen-
dicular to the plate, and the ﬂow velocity is represented by (u(y), 0)
in the (x, y) cartesian coordinate frame. The ﬂow ﬁeld represented
as (u(y), 0) is called a parallel shear ﬂow. Owing to this shear ﬂow,
the plate is acted on by a friction force due to the ﬂow. If the friction
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2.5. An ideal ﬂuid and Newtonian viscous ﬂuid 25
force per unit area of the plate is represented as
σ
f
= µ
du
dy

y=0
, (2.10)
where µ is the coeﬃcient of shear viscosity, this is called the Newton’s
law of viscous friction (see Problem 2.1).
This law can be extended to the law on an internal imaginary
surface of the ﬂow. Consider an internal surface element B perpen-
dicular to the y-axis located at an arbitrary y position (Fig. 2.5).
The unit normal to the surface B is in the positive y direction. The
internal friction force on B from the upper to lower side has only the
x-component. If the friction σ
(s)
per unit area is written as
σ
(s)
= µ
d
dy
u(y), (= q
xy
), (2.11)
then the ﬂuid is called the Newtonian ﬂuid. The friction σ
(s)
per unit
area is called the viscous stress, in particular, called the shear stress
for the present shear ﬂow, and it corresponds to q
xy
of (2.7). The
stress is also termed as a surface force. The pressure force given by
(2.7) and (2.8) in the previous section is another surface force. The
pressure stress has only the normal component to the surface δA(n),
whereas the viscous stress has a tangential component and a normal
component (in general compressible case).
One can consider an idealized ﬂuid in which the shear viscosity µ
vanishes everywhere. Such a ﬂuid is called an inviscid ﬂuid, or an ideal
ﬂuid. In the ﬂow of an inviscid ﬂuid, the velocity adjacent to the solid
wall does not vanish in general, and the ﬂuid has nonzero tangential
u(y)
n = (0,1,0)
B
y
Fig. 2.5. Momentum transfer through an internal surface B.
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26 Fluids
slip-velocity at the wall. On the other hand, the ﬂow velocity of a
viscous ﬂuid vanishes at the solid wall. This is termed as no-slip.
Thus, the boundary conditions of the velocity v on the surface
of a body at rest are summarized as follows:
Viscous ﬂuid: v = 0 (no-slip), (2.12)
Inviscid ﬂuid: nonzero tangential velocity (slip-ﬂow). (2.13)
The inviscid ﬂuid is often called an ideal ﬂuid (or sometimes a perfect
ﬂuid), in which the surface force has only normal component.
In this textbook, the ideal ﬂuid denotes a ﬂuid characterized by
the property that all transport coeﬃcients of viscosity and thermal
conductivity vanish. Since all the transport coeﬃcients vanish, macro-
scopic ﬂows of an ideal ﬂuid is separated from the microscopic irre-
versible dissipative eﬀect arising from atomic thermal motion.
2.6. Viscous stress
For a Newtonian ﬂuid, the viscous stress is given in general by
σ
(v)
ij
= 2µD
ij
+ ζDδ
ij
, (2.14)
where µ and ζ are coeﬃcients of viscosity, and
D
ij
:= e
ij
−
1
3
Dδ
ij
=
1
2
(∂
i
v
j
+ ∂
j
v
i
) −
1
3
(∂
k
v
k
) δ
ij
(2.15)
D := ∂
k
v
k
= e
kk
= div v. (2.16)
The tensor D
ij
is readily shown to be traceless. In fact, D
ii
= e
ii
−
1
3
Dδ
ii
= D −
1
3
D · 3 = 0 since δ
ii
= 3 . It may be said that D
ij
is a
deformation tensor associated with a straining motion which keeps
the volume unchanged. The expression (2.14) of the viscous stress
can be derived from a general linear relation between the stress σ
(v)
ij
and the rate-of-strain tensor e
ij
for an isotropic ﬂuid, in which the
number of independent scalar coeﬃcients is only two — µ and ζ (see
Problem 2.4).
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2.6. Viscous stress 27
Substituting the deﬁnitions of D
ij
and e
ij
of (1.18), the Newtonian
viscous stress is given by
σ
(v)
ij
= µ(∂
i
v
j
+ ∂
j
v
i
−(2/3)Dδ
ij
) + ζDδ
ij
, (2.17)
where the coeﬃcient µ is termed the coeﬃcient of shear viscosity,
while ζ the bulk viscosity (or the second viscosity).
If the surface δA(n) is inclined with its normal n = (n
x
, n
y
, n
z
),
the viscous force F
(v)
i
acting on δA(n) at x from side I (to which the
normal n is directed) to II is given by
F
(v)
i
δA(n) = σ
(v)
ij
n
j
δA(n). (2.18)
For each component, we have
F
(v)
x
= σ
(v)
xx
n
x
+ σ
(v)
xy
n
y
+ σ
(v)
xz
n
x
z,
F
(v)
y
= σ
(v)
yx
n
x
+ σ
(v)
yy
n
y
+ σ
(v)
yz
n
x
z,
F
(v)
z
= σ
(v)
zx
n
x
+ σ
(v)
zy
n
y
+ σ
(v)
zz
n
x
z.
Example 1. Parallel shear ﬂow. Let us consider a parallel shear ﬂow
with velocity v = (u(y), 0, 0). We immediately obtain D = div v =
∂u/∂x = 0. Moreover, all the components of the tensors D
ij
vanish
except D
xy
= D
yx
=
1
2
u

dξ. (2.26)
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30 Fluids
Problem 2.4 Symmetry of stress tensor
Suppose that a stress tensor σ
ij
is acting on the surface of an inﬁnites-
imal cubic volume of side a (with its edges parallel to the axes x, y, z)
in a ﬂuid of density ρ. Considering the balance equation of angular
momentum for the cube and taking the limit as a → 0, verify that
the stress tensor must be symmetric:
σ
ij
= σ
ji
.
Problem 2.5 Stress and strain
Consider a ﬂow ﬁeld v
k
(x) of an isotropic viscous ﬂuid, and suppose
that there is a general linear relation between the viscous stress σ
(v)
ij
and the rate-of-strain tensor e
kl
:
σ
(v)
ij
= A
ijkl
e
kl
.
where e
kl
is deﬁned by (1.18). In an isotropic ﬂuid, the coeﬃcients
A
ijkl
of the fourth-order tensor are represented in terms of isotropic
tensors (see (2.8)), which are given as follows:
A
ijkl
= a δ
ij
δ
kl
+ b δ
ik
δ
jl
+ c δ
il
δ
jk
, (2.27)
where a, b, c are constants. Using this form and the symmetry of
the stress tensor (Problem 2.3), derive the expression (2.14) for the
viscous stress σ
(v)
ij
:
σ
(v)
ij
= 2µD
ij
+ ζDδ
ij
.
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Chapter 3
Fundamental equations
of ideal ﬂuids
Fluid ﬂows are represented by ﬁelds such as the velocity ﬁeld v(x, t),
pressure ﬁeld p(x, t), density ﬁeld ρ(x, t), temperature ﬁeld T(x, t),
and so on. The ﬁeld variables denote their values at a point x and at
a time t. The position vector x is represented by (x, y, z), or equiva-
lently (x
1
, x
2
, x
3
) in the cartesian frame of reference. Fluid particles
move about in the space with a velocity dx/dt = v(x, t).
Flow ﬁeld evolves with time according to fundamental conserva-
tion laws of physics. There are three kinds of conservation laws of
mechancis, which are conservation of mass, momentum and energy.
1
In ﬂuid mechanics, these conservation laws are represented in terms
of ﬁeld variables such as v, p, ρ, etc. Since the ﬁeld variables depend
on (x, y, z) and t, the governing equations are of the form of partial
diﬀerential equations.
1
The conservation laws in mechanics result from the fundamental homogeneity
and isotropy of space and time. From the homogeneity of time in the Lagrangian
function of a closed system, the conservation of energy is derived. From the homo-
geneity of space, the conservation of momentum is derived. Conservation of angu-
lar momentum (which is not discussed in this chapter) results from isotropy
of space. Conservation of mass results from the invariance of the relativistic
Lagrangian in the Newtonian limit (Appendix F.1). See Chap. 12, or [LL75],
[LL76].
31
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32 Fundamental equations of ideal ﬂuids
3.1. Mass conservation
The law of mass conservation is represented by the following Euler’s
equation of continuity, which reads
∂ρ
∂t
+
∂(ρu)
∂x
+
∂(ρv)
∂y
+
∂(ρw)
∂z
= 0, (3.1)
where ρ is the ﬂuid density and v(x, t) = (u, v, w) the velocity. Using
the diﬀerential operator div of the vector analysis, this is written as
∂
t
ρ + div(ρv) = 0, (3.2)
and derived in the following way.
Take a certain volume V
0
ﬁxed in space arbitrarily (Fig. 3.1), and
choose a volume element dV within V
0
. Fluid mass in the volume dV
is given by ρdV , and the total mass is its integral over the volume V
0
,
M
0
(t) =

V
0
ρdV. (3.3)
The ﬂuid density ρ depends on t, and in addition, the ﬂuid itself
moves around with velocity v. Therefore, the total mass M
0
varies
with time t, and its rate of change is given by
d
dt
M
0
(t) =
∂
∂t

V
0
ρdV =

V
0
∂ρ
∂t
dV, (3.4)
where the partial diﬀerential operator ∂/∂t is used on the right-hand
side. Since we are considering ﬁxed volume elements dV in space, the
time deriative ∂
t

ρdV can be replaced by

(∂
t
ρ)dV .
Fig. 3.1. Volume V
0
.
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3.1. Mass conservation 33
The change of total mass M
0
is caused by inﬂow or outﬂow of
ﬂuid through surface A
0
bounding V
0
. The amount of ﬂuid ﬂowing
through a surface element dA with the unit normal n is given by
ρv
n
dA = ρv · ndA,
where v
n
is the normal component of the velocity
2
and v
n
dA denotes
the volume of ﬂuid passing through dA per unit time (Fig. 3.2). The
normal n is taken to be directed outward from V
0
, so that ρv
n
dA
denotes the mass of ﬂuid ﬂowing out of the volume V
0
per unit time.
Total out-ﬂow of the ﬂuid mass per unit time is

A
0
ρ v
n
dA =

A
0
ρv · ndA =

A
0
ρv
k
n
k
dA.
This integral over the closed surface A
0
is transformed into a
volume integral by Gauss’s divergence theorem
3
:

A
0
ρv
k
n
k
dA =

V
0
∂
∂x
k
(ρv
k
) dV =

V
0
div(ρv) dV, (3.5)
where the div operator is deﬁned by (1.31). This gives the rate of
decrease of total mass in V
0
, that is equal to −dM
0
/dt. Thus, on
Fig. 3.2. Mass ﬂux through dA.
2
v
n
= v · n = v
k
n
k
= |v||n| cos θ, where |n| = 1 and θ is the angle between v
and n.
3
The rule for transforming the surface integral into a volume integral with the
bounding surface is as follows: the term n
k
dA in the surface integral is replaced
by the volume element dV and the diﬀerential operator ∂/∂x
k
acting on the
remaining factor in the integrand of the surface integral.
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34 Fundamental equations of ideal ﬂuids
addition of the two terms, the right-hand sides of (3.4) and (3.5),
must vanish:

V
0
∂ρ
∂t
dV +

V
0
div(ρv)dV =

V
0
¸
∂ρ
∂t
+ div(ρv)

dV = 0. (3.6)
This is an identity and must hold for any choice of volume V
0
within the ﬂuid. Hence, the integrand inside [ ] must vanish point-
wise.
4
Thus, we obtain the equation of continuity (3.2), which is also
written as
∂
∂t
ρ +
∂
∂x
k
(ρv
k
) = 0. (3.7)
The second term of (3.7) is decomposed as
∂
∂x
k
(ρv
k
) = v
k
∂
k
ρ + ρ∂
k
v
k
= (v · ∇)ρ + ρ div v,
where ∂
k
= ∂/∂x
k
. Using the Lagrange derivative D/Dt of (1.13),
the continuity equation (3.7) is rewritten as
Dρ
Dt
+ ρ div v = 0. (3.8)
Notes: (i) If the density of each ﬂuid particle is invariant during the
motion, then Dρ/Dt = 0. In this case, the ﬂuid is called incompress-
ible. If the ﬂuid is incompressible, we have
div v = ∂
x
u + ∂
y
v + ∂
z
w = 0. (3.9)
This is valid even when ρ is not uniformly constant.
(ii) Uniform density: The same equation (3.9) is also obtained
from (3.2) by setting ρ to be a constant.
Thus, Eq. (3.9) implies both cases (i) and (ii).
4
Otherwise, the integral does not always vanish, e.g. when choosing V
0
where the
integrand [ ] is not zero.
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3.3. Conservation form 35
3.2. Conservation form
It would be instructive to remark a general characteristic feature of
Eq. (3.7). Namely, it has the following structure:
∂
∂t
D +
∂
∂x
k

F

k
= Q, (3.10)
where D is a density of some physical ﬁeld and (F)
k
is the kth com-
ponent of corresponding ﬂux F, and Q is a source generating the
ﬁeld D (Fig. 3.3). If D is the mass density ρ and (F)
k
is the mass
ﬂux ρv
k
passing through a unit surface per unit time (and there is no
mass source Q = 0), then Eq. (3.10) reduces to the continuity equa-
tion (3.7). The equation of the form (3.10) is called the conservation
form, in general.
3.3. Momentum conservation
The conservation of momentum is the fundamental law of mechan-
ics. This conservation law results from the homogeneity of space
with respect to the Lagrangian function in Newtonian mechanics.
However, we write down ﬁrstly the equation of motion for a ﬂuid
particle of mass ρδV in the form, (ρδV )(acceleration) = (force).
5
Fig. 3.3. Conservation form.
5
This relation is the Newton equation of motion itself for a point mass (a discrete
object). Fundamental equations and conservation equations for continuous ﬁelds
is the subject of Chapter 12 (gauge principle for ﬂows of ideal ﬂuids).
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36 Fundamental equations of ideal ﬂuids
Thereafter, it will be shown that this is equivalent to the conserva-
tion of momentum.
3.3.1. Equation of motion
In an ideal ﬂuid, the surface force is only the pressure (2.8) or (2.9)
(there is no viscous stress). Choosing a volume V
0
as before and
denoting its bounding surface by A
0
, the total pressure force on A
0
is given by
−

A
0
pn
i
dA = −

V
0
∂
∂x
i
pdV = −

V
0
(grad p)
i
dV, (3.11)
where the middle portion is obtained by applying the rule described
in the footnote in Sec. 3.1. By this extended Gauss theorem, the sur-
face integral is transformed to the volume integral. It is remarkable
to ﬁnd that the force on a small ﬂuid particle of volume δV is given
by the pressure gradient,
−grad p δV = −(∂
x
p, ∂
y
p, ∂
z
p) δV.
In addition to the surface force just given, there is usually a volume
force which is proportional to the mass ρδV . Let us write this as
(ρδV ) f , which is often called an external force. A typical example is
the gravity force, (ρδV ) g with g as the acceleration of gravity.
Once the velocity ﬁeld v(x, t) is given, the acceleration of the ﬂuid
particle is written as Dv/Dt (see (1.33)). The Newton equation of
motion for a ﬂuid particle is written as
(ρδV )
D
Dt
v = −grad pδV + (ρδV) f . (3.12)
Dividing this by ρδV and using the expression given in (1.33), we
obtain
∂
t
v + (v · ∇)v = −
1
ρ
grad p +f . (3.13)
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3.3. Momentum conservation 37
This is called Euler’s equation of motion.
6
The second term
(v · ∇)v on the left-hand side is also written as (v · grad)v. This
is of the second order with respect to the velocity vector v, and
often called a nonlinear term, or advection term. The nonlinearity is
responsible for complex behaviors of ﬂows. This term is also called an
inertia term (however, the term ∂
t
v is related to ﬂuid inertia as well).
In the component representation, the ith component of the
equation is
∂
t
v
i
+ v
k
∂
k
v
i
= −
1
ρ
∂
i
p + f
i
. (3.14)
If the external force is the uniform gravity represented by f =
(0, 0, −g) where the acceleration of gravity g is constant and directed
towards the negative z axis, each component is written down as
follows:
∂
t
u + u∂
x
u + v∂
y
u + w∂
z
u = −ρ
−1
∂
x
p,
∂
t
v + u∂
x
v + v∂
y
v + w∂
z
v = −ρ
−1
∂
y
p,
∂
t
w + u∂
x
w + v∂
y
w + w∂
z
w = −ρ
−1
∂
z
p −g.
(3.15)
Euler’s equation of motion is given another form. To see it, the fol-
lowing vector identity (A.21) is useful:
v ×(∇×v) = ∇

1
2
v
2

−(v · ∇)v. (3.16)
6
The paper of Leonhard Euler (1707–1783) was published in the proceedings of
the Royal Academy Prussia (1757) in Berlin with the title, Principes g´en´eraux du
mouvement des ﬂuides (General principles of the motions of ﬂuids). The form of
equation Euler actually wrote down is that using components such as (3.15) (with
x, y components of external force), determined from the principles of mechanics
(as carried out in the main text). Before that, he showed in the same paper the
equation of continuity exactly of the form (3.1). The equation of continuity had
been derived in his earlier paper (Principles of the motions of ﬂuids, 1752) by
assuming that continuity of the ﬂuid is never interrupted. It is rather surprising
to ﬁnd that not only Euler (1757) derived an integral for potential ﬂows of the
form (5.29) of Chap. 5, but also the z-component of the vorticity equation (3.32) of
Sec. 3.4.1 was derived in 1752. Thus, Euler presented an essential part of modern
ﬂuid dynamics of ideal ﬂuids. However, his contribution should be regarded as one
step toward the end of a long process, carried out by Bernoulli family, d’Alembert,
Lagrange, etc. in the 18th century [Dar05].
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38 Fundamental equations of ideal ﬂuids
Eliminating the advection term (v · ∇)v in (3.13) with the help
of (3.16) and introducing the deﬁnition ω := ∇ × v called the vor-
ticity, the equation of motion (3.13) is rewritten as
∂
t
v +ω ×v = −∇
v
2
2
−
1
ρ
∇p +f . (3.17)
Consider a particular case when the density is uniform, i.e.
ρ = ρ
0
(constant), and that the external force has a potential χ,
i.e. f = −∇χ. Then the above equation reduces to
∂
t
v +ω ×v = −∇

v
2
2
+
1
ρ
0
p + χ

. (3.18)
The gravity potential of uniform acceleration g is given by
χ = gz. (3.19)
3.3.2. Momentum ﬂux
Euler’s equation of motion just derived represents the conservation
of momentum. In fact, consider the momentum included in the vol-
ume element δV , which is given by (ρδV )v. Therefore, ρv is the
momentum in a unit volume, namely the momentum density. Its
time derivative is
∂
t
(ρv
i
) = (∂
t
ρ)v
i
+ ρ∂
t
v
i
.
The two time derivatives ∂
t
ρ and ∂
t
v
i
can be eliminated by using
(3.7) and (3.14), and we obtain
∂
t
(ρv
i
) = −∂
k
(ρv
k
)v
i
−ρv
k
∂
k
v
i
−∂
i
p + ρf
i
= −∂
k
(ρv
k
v
i
) −∂
i
p + ρf
i
. (3.20)
The pressure gradient term on the right-hand side is rewritten as
∂
i
p = ∂
k
(pδ
ik
),
by using Kronecker’s delta δ
ij
(A.1). Substituting this and rearrang-
ing (3.20), we obtain
∂
t
(ρv
i
) + ∂
k
P
ik
= ρf
i
, (3.21)
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3.3. Momentum conservation 39
where the tensor P
ik
is deﬁned by
P
ik
= ρv
i
v
k
+ pδ
ik
. (3.22)
Equation (3.21) represents the conservation of momentum, which
is seen to have the form of Eq. (3.10). The quantity ρv
i
is the ith
component of the momentum density (vector), while P
ik
denotes the
momentum ﬂux (tensor).
7
The diﬀerence from the previous equation
of mass conservation is that there is a source term ρf
i
on the right-
hand side. This is natural because the external force is a source of
momentum in the true meaning.
Why is the term on the right-hand side of (3.21) a source (produc-
tion) of momentum will become clearer, if we integrate the equation
over a volume V
0
. In fact, we have
∂
∂t

V
0
ρv
i
dV +

V
0
∂
∂x
k
P
ik
dV =

V
0
ρf
i
dV.
The second integral term on the left-hand side can be transformed to
a surface integral by using the rule of the Gauss theorem, i.e. replace
∂
k
(X) dV with Xn
k
dA. Moving this term to the right-hand side, we
obtain
∂
∂t

V
0
ρv
i
dV = −

A
0
P
ik
n
k
dA +

V
0
ρf
i
dV. (3.23)
This is interpreted as follows. The left-hand side denotes the rate
of change of total momentum included in volume V
0
, whereas the
ﬁrst integral on the right-hand side represents the momentum ﬂow-
ing out of the bounding surface A
0
per unit time, and the second
integral is the total force acting on V
0
which is nothing but the rate
of production of momentum per unit time in mechanics. Thus, the
Euler equation of motion represents the momentum conservation in
conjunction with the continuity equation.
7
The mass density was a scalar, whereas the momentum density is a vector.
Correspondingly, the ﬂux P
ik
is a second-order tensor.
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40 Fundamental equations of ideal ﬂuids
An important point brought to light is the following expression,
P
ik
n
k
dA = ρv
i
v
k
n
k
dA + pδ
ik
n
k
dA, (3.24)
where (3.22) is used. This represents the ith component of total
momentum ﬂux passing through the surface element dA. The ﬁrst
term is the macroscopic momentum ﬂux, whereas the second rep-
resents a microscopic momentum ﬂux. In Sec. 1.2 we saw that the
microscopic momentum ﬂux gives the expression for pressure of an
ideal gas.
3.4. Energy conservation
The conservation of energy is a fundamental law of physics. An ideal
ﬂuid is characterized by the absence of viscosity and thermal dif-
fusivity. Because of this property, the motion of a ﬂuid particle is
adiabatic. It will be shown that this is consistent with the conserva-
tion of energy.
3.4.1. Adiabatic motion
In an ideal ﬂuid, there is neither heat generation by viscosity nor heat
exchange between neighboring ﬂuid particles. Therefore the motion
of an ideal ﬂuid is adiabatic, and the entropy of each ﬂuid particle
is invariant during its motion. Denoting the entropy per unit mass
by s, the adiabatic motion of a ﬂuid particle is described as
D
Dt
s = ∂
t
s + (v · ∇)s = 0. (3.25)
It is said, the motion is isentropic. This equation is transformed to
∂
t
(ρs) + div(ρsv) = 0, (3.26)
by using the continuity equation (3.2). If the entropy value was
uniform initially, the value is invariant thereafter. This is called a
homentropic ﬂow. Then the equations will be simpliﬁed.
Denoting the internal energy and enthalpy per unit mass by e
and h, respectively and the speciﬁc volume by V = 1/ρ, we have a
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3.4. Energy conservation 41
thermodynamic relation h = e + pV . For an inﬁnitesimal change of
state, two thermodynamic relations are written down as
de = Tds −pdV = Tds + (p/ρ
2
) dρ, (3.27)
dh = Tds + V dp = Tds + (1/ρ) dp, (3.28)
[LL80, Chap. 2], where T is the temperature.
For a homentropic ﬂuid motion, we have ds = 0, and therefore
dh = dp/ρ. Thus, we obtain
(1/ρ) grad p = grad h.
Substituting this into (3.13) and assuming the external force is con-
servative, i.e. f = −grad χ, we obtain
∂
t
v + (v · ∇)v = −gradh −grad χ. (3.29)
Equation (3.17) reduces to
∂
t
v +ω ×v = −grad

v
2
2
+ h + χ

. (3.30)
This is valid for compressible ﬂows, while Eq. (3.18) is restricted to
the case of ρ = const. Taking the curl of this equation, the right-hand
side vanishes due to (A.25) of Appendix A.5, and we obtain
∂
t
ω + curl(ω ×v) = 0, ω = curl v. (3.31)
Thus, we have obtained an equation for the vorticity ω, which
includes only v since ω = curl v (p and other variables are elim-
inated). In this equation, the entropy s is assumed a constant,
although ρ is variable.
Using (A.24) of Appendix A.4, this is rewritten as
∂
t
ω + (v · grad)ω + (div v)ω = (ω · grad)v. (3.32)
Note that the ﬁrst two terms are uniﬁed to Dω/Dt and the third
term is written as −(1/ρ)(Dρ/Dt)ω by (3.8). Dividing (3.32) by ρ,
1
ρ
Dω
Dt
−
ω
ρ
2
Dρ
Dt
=
1
ρ
(ω · grad)v.
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42 Fundamental equations of ideal ﬂuids
This is transformed to a compact form:
D
Dt

ω
ρ

=

ω
ρ
· grad

v. (3.33)
3.4.2. Energy ﬂux
Let us consider an energy ﬂux just as the momentum ﬂux in
Sec. 3.3.2. Total energy in a volume element δV is given by the sum
of the kinetic and internal energies as
ρδV

= W, (3.37)
where W = ρv
k
f
k
denotes the rate of work by the external force.
This is the equation of conservation of energy in the form of (3.10).
When there is no external force, the right-hand side W vanishes
(Fig. 3.4).
It is remarked that the energy density is given by the sum of
kinetic energy and internal energies, whereas the energy ﬂux (F)
k
is
(F)
k
= ρv
k

1
2
v
2
+ h

,
namely e is replaced by h for the ﬂux F.
Integrating (3.37) over volume V
0
and transforming the volume
integral of the second term into the surface integral, we have
∂
t

V
0

1
2
ρv
2
+ ρe

dV = −

A
0

1
2
ρv
2
+ ρe + p

v
k
n
k
dA
+

V
0
WdV. (3.38)
This indicates that the rate of increase of the total energy in V
0
is
given by the sum of the energy inﬂow −

1
2
ρv
2
+ ρe

v
k
n
k
through
the bounding surface and the rate of work by the pressure −pv
k
n
k
,
in addition to the rate of work W by the external force.
pv
n
energy density
energy flux
Fig. 3.4. Energy density E =
1
2
v
2
+ e, and energy ﬂux Q
n
= ρEv
n
+ pv
n
=
ρ(
1
2
v
2
+ h)v
n
.
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44 Fundamental equations of ideal ﬂuids
3.5. Problems
Problem 3.1 One-dimensional unsteady ﬂow
Write down the three conservation equations of mass, momentum
and energy for one-dimensional unsteady ﬂows (in the absence of
external force) with x as the spatial coordinate and t the time when
the density is ρ(x, t), velocity is v = (u(x, t), 0, 0), and so on.
Comment: Lagrange’s form of equation of motion
Lagrangian representation of position of a ﬂuid particle a = (a, b, c)
at time t is deﬁned by X(a, t) in (1.11) with its velocity V
a
(t) =
∂
t
X(a, t). Then, the x component of equation of motion (3.12) can
be written as ∂
2
t
X(a, t) = −∂
x
p + f
x
. We multiply this by ∂x/∂a.
Similarly, multiplying y and z components of Eq. (3.12) and summing
up the three expressions, we obtain
∂
2
t
X·
∂X
∂a
= −
∂p
∂a
+f ·
∂X
∂a
. (3.39)
This is the Lagrange’s form of equation of motion (1788), correspond-
ing to the Eulerian version (3.13).
Deﬁning the Lagrangian coordinates a = (a
i
) by the particle posi-
tion at t = 0, the continuity equation for the particle density ρ
a
(t) is
given by
ρ
a
(t)
∂(x)
∂(a)
= ρ
a
(0), (3.40)
where ∂(x)/∂(a) =det(∂X
j
/∂a
i
) is the Jacobian determinant.
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Chapter 4
Viscous ﬂuids
4.1. Equation of motion of a viscous ﬂuid
In the previous chapters, we learned the existence of surface forces to
describe ﬂuid motion in addition to the volume force such as gravity.
The surface force is also termed as the stress and represented by a
tensor. A typical stress is the pressure pδ
ij
,
1
which is written as
σ
(p)
ij
= −pδ
ij
= −

¸
p 0 0
0 p 0
0 0 p
¸

. (4.1)
The pressure force acts perpendicularly to a surface element δA(n)
and is represented as −pn
i
dA (see (2.8), (2.9)). Another stress was
the viscous stress (internal friction) σ
(v)
ij
considered in Sec. 2.6, which
has a tangential force to the surface δA(n) as well. A typical one is
the shear stress in the parallel shear ﬂow in Sec. 2.5.
The conservation of momentum of ﬂows of an ideal ﬂuid is given
by (3.21) and (3.22), which reads
∂
t
(ρv
i
) +∂
k
P
ik
= ρf
i
, (4.2)
where P
ik
= ρv
i
v
k
+ pδ
ik
is the momentum ﬂux tensor for an ideal-
ﬂuid ﬂow. In order to obtain the equation of motion of a viscous ﬂuid,
a viscous stress should be added to the “ideal” momentum ﬂux P
ik
,
1
The tensor of the form δ
ij
is called an isotropic tensor. For any vector A
i
, the
transformation δ
ij
A
j
is A
i
.
45
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46 Viscous ﬂuids
which is written as −σ
(v)
ij
representing irreversible viscous transfer of
momentum.
2
Thus we write the momentum ﬂux tensor in a viscous
ﬂuid in the form,
P
ik
= ρv
i
v
k
+p
ik
−σ
(v)
ik
, (4.3)
σ
ik
= −pδ
ik
+σ
(v)
ik
, (4.4)
where σ
ij
is the stress tensor written as q
ij
in Sec. 2.4, and σ
(v)
ij
the
viscous stress tensor, deﬁned by (2.17) for a Newtonian ﬂuid.
Substituting this in (4.2) and eliminating some terms by using the
continuity equation, we obtain the equation of motion of a Newtonian
viscous ﬂuid. The result is that a new term of the form ∂σ
(v)
ik
/∂x
k
is
added to the right-hand side of the Euler equation (3.14):
ρ(∂
t
v
i
+v
k
∂
k
v
i
) = −∂
i
p +
∂
∂x
k
σ
(v)
ik
, (4.5)
where the external force f
i
is omitted.
If the viscosity coeﬃcients µ and ζ are regarded as constants,
3
the viscous force is written as
∂
∂x
k
σ
(v)
ik
= µ

= W,
(4.15)
where the heat ﬂux q = −k grad T = (−k∂
j
T) is added to the energy
ﬂux in addition to the energy transfer −v
i
σ
(v)
ik
due to microscopic
processes of internal friction.
For consistency with thermodynamic relations as well as the con-
tinuity equation and equation of motion (Navier–Stokes equation),
the following equation for the entropy s per unit mass is required to
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4.3. Energy dissipation in an incompressible ﬂuid 49
hold [LL87, Sec. 49]:
ρT(∂
t
s + (v ∇)s) = σ
(v)
ik
∂
k
v
i
+ div(k grad T). (4.16)
This is a general equation of heat transfer (see (9.36) for the equa-
tion of thermal conduction). In the entropy equation (3.25) of an
ideal ﬂuid, there was no term on the right-hand side, resulting in
the conservation of entropy. The expression on the left ρT(Ds/Dt)
is the quantity of heat gained per unit volume in unit time. There-
fore, the expressions on the right denote the heat production due to
viscous dissipation of energy and thermal conduction.
A deeper insight into the entropy equation will be gained if we
consider the rate of change of total entropy

ρs dV in a volume V
0
.
In fact, using Eq. (4.16) and the continuity equation, one can derive
the following equation:
d
dt

V
0
ρsdV =

µ
2T
(∂
k
v
i
+∂
i
v
k
−(2/3)(div v)δ
ik
)
2
dV
+

ζ
T
(div v)
2
dV +

k
(grad T)
2
T
2
dV (4.17)
[LL87, Sec. 49]. The entropy can only increase. Namely, each integral
term on the right must be positive. The ﬁrst two terms are the rate
of entropy production due to internal friction, while the last term is
that owing to thermal conduction. Hence it follows that the viscosity
coeﬃcients µ and ζ must be positive as well as the thermal conduction
coeﬃcient k.
4.3. Energy dissipation in an incompressible ﬂuid
In an incompressible viscous ﬂuid of uniform density ρ
0
where
div v = 0, the energy equation becomes simpler and clearer. Mul-
tiplying v
i
to the Navier–Stokes equation (4.5), one has
ρ
0
(v
i
∂
t
v
i
+v
i
v
k
∂
k
v
i
) = −v
i
∂
i
p +v
i
∂
k
σ
(v)
ik
. (4.18)
On the left, we can rewrite as v
i
∂
t
v
i
= ∂
t

, (4.20)
where ν = µ/ρ
0
.
Suppose that the ﬂow space V
∞
is unbounded and the velocity
decays suﬃciently rapidly at inﬁnity. Taking the integral of Eq. (4.20)
over a suﬃciently large bounded space V and transforming the vol-
ume integral on the right to an integral over the surface S
∞
at large
distance (which recedes to inﬁnity later), we obtain
∂
t

V
1
2
v
2
dV +ν

V
(∂
k
v
i
)
2
dV
=

S
∞

−v
k
1
2
v
2
−v
k
p +νv
i
∂
k
v
i

n
k
dS.
It is assumed that the velocity v
k
decays suﬃciently rapidly and the
surface integral vanishes in the limit when it recedes to inﬁnity.
5
Thus
we obtain the equation of energy decay of ﬂows of a viscous ﬂuid:
dK
dt
= −ν

V
[ω[
2
dV, (4.24)
since the integrated surface terms vanish.
4.4. Reynolds similarity law
In studying the motions of a viscous ﬂuid, the ﬂow ﬁelds are often
characterized with a representative velocity U and length L. For
example (Fig. 4.1), one can consider a spherical ball of diameter L
moving with a velocity U. Another case is a viscous ﬂow in a circular
pipe of diameter L with a maximum ﬂow speed U.
Fig. 4.1. Representative scales.
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52 Viscous ﬂuids
When a ﬂow of an incompressible viscous ﬂuid of density ρ
0
has a
single representative velocity U and a single representative length L,
the state of ﬂow is characterized by a single dimensionless number,
deﬁned by
R
e
=
UL
ν
=
ρ
0
UL
µ
, (4.25)
which is called the Reynolds number. This is formulated as
follows.
Suppose that the space coordinates x = (x, y, z), time t, velocity v
and pressure p are normalized to dimensionless variables. Denoting
the dimensionless variables by primes, we deﬁne
x

=
x
L
, t

=
t
τ
, v

=
v
U
, p

=
p −p
0
ρ
0
U
2
,
where the normalization is done using L as the dimension of length,
U for the dimension of velocity, τ = L/U for time, and U/τ = U
2
/L
for acceleration. The dimension of pressure is the same as that of
[pressure] = [force]/[area] = ([mass] [acceleration])/[area].
The representative value of pressure variation is of the order of
[(ρ
0
L
3
)(U
2
/L)/L
2
] = [ρ
0
U
2
], with the reference pressure denoted
by p
0
.
Substituting these into (4.8) (without the external force f ) and
rewriting it with primed variables, we obtain
∂
∂t

v

+ (v

∇

)v

= −∇

p

+
1
R
e
(∇

)
2
v

(4.26)
(∇

v

= 0). Equation (4.26) thus derived is a dimensionless equation
including a single dimensionless constant R
e
(Reynolds number). If
the values of R
e
are the same between two ﬂows under the same
boundary condition, the ﬂow ﬁelds represented by the two solutions
are equivalent, even though the sets of values U, L, ν are diﬀer-
ent. This is called the Reynolds similarity law, and R
e
is termed the
similarity parameter.
Stating it in another way, if the values of R
e
are diﬀerent, the
corresponding ﬂow ﬁelds are diﬀerent, even though the boundary
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4.4. Reynolds similarity law 53
conditions are the same. From this point of view, R
e
is also termed
as the control parameter.
Consider a steady ﬂow in which all the ﬁeld variables do not
depend on time and hence the ﬁrst term of (4.8) vanishes. Let us
estimate relative magnitude of the second inertia term. Estimating
the magnitude of velocity gradient ∇v to be of the order of U/L, the
magnitude of the second term of (4.8) is [(v ∇)v[ = O(U(U/L)) =
O(U
2
/L). Regarding the last viscous term, analogous estimation
leads to [ν∇
2
v[ = O(νU/L
2
). Taking the ratio of the two terms,
we obtain
O([(v ∇)v[)
O([ν∇
2
v[)
=
U
2
/L
νU/L
2
=
UL
ν
= R
e
,
which is nothing but the Reynolds number. Note that the dimension
of ν is the same as that of UL = [L]
2
[T]
−1
with [T] the dimension
of time.
If the ﬂuid viscosity is high and its kinematic viscosity ν is large
enough, the viscous term [ν∇
2
v[ will be larger than the inertia term
[(v ∇)v[, resulting in R
e
< 1. Such a ﬂow of low Reynolds number is
called a viscous ﬂow, including the case of R
e
· 1 (see Sec. 4.8). The
ﬂow of R
e
< 1 is said to be a slow motion, because a ﬂow of very small
velocity U makes R
e
< 1 regardless of the magnitude of viscosity.
The motion of a microscopic particle becomes inevitably a ﬂow of
low Reynolds number because the length L is suﬃciently small.
On the contrary, we will have R
e
1 for ﬂows of large U, large L,
or small ν. Such a ﬂow is said to be a ﬂow of high Reynolds number.
In the ﬂows of high Reynolds numbers, the magnitude of inertia term
[(v ∇)v[ is much larger than the viscous term [ν∇
2
v[, and the ﬂow
is mainly governed by the ﬂuid inertia. However, it will be found in
the next section that the role of viscosity is still important in the
ﬂows of high Reynolds numbers too. Most ﬂows at high Reynolds
numbers become turbulent.
Smooth ﬂows observed at low Reynolds numbers are said to be
laminar. Consider a sequence of states as the value of Reynolds
number R
e
is increased gradually from a low value at the state of a
laminar ﬂow. According to the increase in Reynolds number, the ﬂow
varies its state, and at suﬃciently large values of R
e
, it will change
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54 Viscous ﬂuids
over to a turbulent state. This transition to turbulence occurs at
a ﬁxed value of Reynolds number, which is termed as the critical
Reynolds number, often written as R
c
.
4.5. Boundary layer
As the Reynolds number R
e
is increased, the viscous term takes small
values over most part of the space of ﬂow ﬁeld, and the viscous action
tends to be localized in space. But, however large the value of R
e
, the
viscosity eﬀect can never disappear. This fact means that the ﬂow
in the limit of vanishing viscosity ν (i.e. ν →0) does not necessarily
coincide with the ﬂow of inviscid ﬂow (ν = 0). As an example, we
consider a plane boundary layer ﬂow.
6
Suppose that there is a ﬂow of an incompressible viscous ﬂuid over
a plane wall AB and the ﬂow tends to a uniform ﬂow of velocity U
far from the wall, and that the ﬂow is steady (Fig. 4.2). We take the
x axis in the direction of ﬂow along the wall AB and the y axis per-
pendicular to AB. The ﬂow can be described in the two-dimensional
(x, y) space. The velocity becomes zero on the wall y = 0 by the
no-slip condition. Velocity distributions with respect to the coordi-
nate y are schematically represented in the ﬁgure. In this situation,
the change of velocity occurs in a thin layer of thickness δ (say)
adjacent to the wall, and a boundary layer is formed. The boundary
y
U(A)
U(B)
l
O
A
B
x
Fig. 4.2. Plane boundary layer.
6
Plane in ﬂuid mechanics means “two-dimensional”.
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4.5. Boundary layer 55
conditions are summarized as follows:
(u, v) →(U, 0) as y/δ →∞, (4.27)
(u, v) = (0, 0) at y = 0. (4.28)
Provided that the velocity ﬁeld is expressed as (u, v) with the
kinematic viscosity ν and the uniform density ρ, the Navier–Stokes
equation (4.8) in the two-dimensional (x, y) space reduces to
u
t
+uu
x
+vu
y
= −(1/ρ) p
x
+ν(u
xx
+u
yy
), (4.29)
v
t
+uv
x
+vv
y
= −(1/ρ) p
y
+ν(v
xx
+v
yy
), (4.30)
u
x
+v
y
= 0, (4.31)
where the ﬁrst time derivative terms of (4.29) and (4.30) vanish in the
steady problem under consideration. Within the boundary layer, the
change in the y direction is more rapid compared with the change in
the x direction. Hence, the term νu
xx
may be much smaller than the
term νu
yy
in Eq. (4.29), and hence the term νu
xx
may be neglected.
But, there is a change in the x direction however slight. The two terms
u
x
and v
y
in the continuity equation (4.31) balance each other.
On the basis of the above estimates and the estimates below (4.33),
the steady ﬂow in a boundary layer can be well described asymptoti-
cally by the following systemof equations in the limit of small viscosity
ν (Prandtl (1904); see Problem 4.4 for Blasius ﬂow and Problem 4.5):
uu
x
+vu
y
= −(1/ρ) p
x
+νu
yy
, (4.32)
u
x
+v
y
= 0. (4.33)
These are consistent with the estimate of the order of magnitude just
below. It is useful to recognize that the scales of variation are diﬀerent
in the two directions x and y. Suppose that the representative scales
in the directions x and y are denoted by l and δ, respectively where
l δ, and the magnitude of u is given by U. The viscous term on the
right-hand side of Eq. (4.32) can be estimated as the order O(νU/δ
2
).
Both the ﬁrst and second terms on the left-hand side are estimated
as the order O(U
2
/l). Equating the above two estimates in the order
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56 Viscous ﬂuids
of magnitude, we have
U
2
l
=
νU
δ
2
, or
δ
l
=

ν
Ul
=
1
√
R
e
, (4.34)
where R
e
= Ul/ν. It is seen that the thickness δ of the boundary
layer becomes smaller, as the Reynolds number R
e
increases. From
the above expression, one ﬁnds the behavior δ ∝
√
l, a parabolic
growth of the thickness δ along the wall. The boundary layer does
not disappear, however small the viscosity ν is.
More importantly, in the boundary layer there exists nonzero vor-
ticity ω. This is seen from the fact that ω = v
x
−u
y
≈ −u
y
(nonzero),
where v
x
is very small. Far from the wall, the ﬂow velocity tends to a
uniform value and therefore ω → 0 as y → ∞. This implies that the
wall contributes to the generation of vorticity. An important diﬀerence
of the ﬂow at high Reynolds numbers from the ﬂow of an inviscid ﬂuid
is the existence of such a rotational layer at the boundary. In fact, this
is an essential diﬀerence from the inviscid ﬂow.
4.6. Parallel shear ﬂows
Consider a simple class of ﬂows of a viscous ﬂuid having only x
component u of velocity v:
v = (u(y, z, t), 0, 0). (4.35)
The continuity equation reduces to the simple form ∂
x
u = 0, stating
that u is independent of x, and consistent with (4.35). This is called
the parallel shear ﬂow, or unidirectional ﬂow. In this type of ﬂows,
the convection term vanishes identically since (v ∇)u = u∂
x
u = 0.
Without the external force f , the x, y, z components of the Navier–
Stokes equation (4.8) reduce to
∂
t
u −ν(∂
2
y
+∂
2
z
)u = −
1
ρ
0
∂
x
p (4.36)
∂
y
p = 0, ∂
z
p = 0. (4.37)
From the last equations, the pressure should be a function of x and t
only: p = p(x, t). The left-hand side of (4.36) depends on y, z, t, while
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4.6. Parallel shear ﬂows 57
the right-hand side depends on x, t. Therefore, the equality states
that both sides should be a function of t only for the consistency of
the equation. Writing it as P(t)/ρ
0
, we have
−grad p = (P, 0, 0), (4.38)
in which the ﬂuid is driven to the positive x direction when P is
positive.
4.6.1. Steady ﬂows
In steady ﬂows, we have ∂
t
u = 0 and P = const. The above equation
(4.36) reduces to
∂
2
y
u +∂
2
z
u = −
P
µ
, (4.39)
where µ is the dynamic viscosity. The acceleration of a ﬂuid particle
vanishes identically in steady unidirectional ﬂows, hence the density
does not make its appearance explicitly.
A solution u = u(y, z) satisfying (4.39) is in fact an exact solution
of the incompressible Navier–Stokes equation. Some of such solutions
are as follows :
u
C
(y) =
U
2b
(y +b), P = 0, ([y[ < b) : Couette ﬂow, (4.40)
u
P
(y) =
P
2µ
(b
2
−y
2
), P = 0, ([y[ < b) : 2D Poiseuille ﬂow.
(4.41)
Obviously, these satisfy Eq. (4.39). We have another axisymmet-
ric solution, called the Hagen–Poiseuille ﬂow which is considered as
Problem 4.1 at the end of this chapter.
The ﬁrst Couette ﬂow represents a ﬂow between the ﬁxed plate
at y = −b and the plate at y = b moving with velocity U in the x
direction when there is no pressure gradient [Fig. 4.3(a)].
The second is the Poiseuille ﬂow [Fig. 4.3(b)] between two parallel
walls at y = ±b under a constant pressure gradient P (d = 2b in
(2.22)). The maximum velocity U is attained at the center y = 0,
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58 Viscous ﬂuids
U
b b
P U
U
−b −b
b
−b
P
Fig. 4.3. (a) Couette ﬂow, (b) Poiseuille ﬂow, and (c) combined ﬂow.
given by
U = Pb
2
/(2µ). (4.42)
The total rate of ﬂow Q per unit time is
Q =

, (4.53)
0 = −
1
ρ
0
∂
z
p.
From the last equation, we have p = p(r, θ, t). However, the pressure
p must be of the form p = p(r, t). Otherwise, ∂
θ
p is a function of r and
t from the second equation of (4.53), and the pressure p becomes a
multivalued function in general with respect to the azimuthal angle θ,
i.e. p(θ, t) = p(θ + 2π, t), which is not permissible.
Assuming p = p(r, t), the second equation of (4.53) becomes
∂
t
v
θ
= ν

δ(x) δ(y) δ(z). (4.66)
Namely, the Stokeslet is a particular solution of the Stokes equation
and represents a creeping ﬂow of a viscous ﬂuid subject to a concen-
trated external force of magnitude 8πµ acting at the origin in the
positive x direction.
4.8.3. Slow motion of a sphere
Consider a steady slow motion of a very small spherical particle of
radius a in a viscous ﬂuid with a constant velocity −U. Let us observe
this motion relative to the reference frame F
C
ﬁxed to the center of
the sphere. Fluid motion observed in the frame F
C
would be a steady
ﬂow around a sphere of radius a, tending to a uniform ﬂow of velocity
U at inﬁnity. Then, the ﬂow ﬁeld is axisymmetric and described by
the Stokes’s stream function Ψ (Appendix B.3). Such a ﬂow can be
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66 Viscous ﬂuids
represented as
Ψ
Us
(r, θ) =
1
2
Ur
2
sin
2
θ +
1
4
U
a
3
r
sin
2
θ −
3
4
UaΨ
S
(r, θ). (4.67)
Each of the three terms is a solution of the Stokes equation (4.60).
In fact, the third term is the Stokeslet considered in the previous
subsection and a solution of (4.60) with a singularity at the origin.
The ﬁrst term represents a uniform ﬂow of velocity U in the x direc-
tion.
9
The second term is a dipole in the positive x direction. These
are equivalent to the stream functions given by (5.43) and (5.44) in
Chapter 5. As described in detail there, the ﬁrst two terms repre-
sent irrotational incompressible ﬂows. Hence they are solutions of
the Stokes equation as remarked at the end of Sec. 4.8.1.
Thus, the above stream function Ψ
Us
is a solution of the Stokes
equation as a whole as well because of the linearity of Eq. (4.60).
10
It
is to be remarked that the stream function is of the form Uf (r) sin
2
θ.
This manifests that the stream lines are rotationally symmetric with
respect to the polar axis x, and in addition, has a symmetry sin
2
θ =
sin
2
(π −θ), i.e. a left–right symmetry or fore–aft symmetry.
Substituting (4.67) into (4.61), we obtain
v
r
= U

1 −
3
2
a
r
+
1
2
a
3
r
3

cos θ, (4.68)
v
θ
= −U

1 −
3
4
a
r
−
1
4
a
3
r
3

sin θ. (4.69)
At r = a, we have v
r
= 0, v
θ
= 0. Hence the no-slip condition
is satisﬁed on the surface of the sphere of radius a. In addition, at
inﬁnity the velocity tends to the uniform ﬂow:
(v
r
, v
θ
) →U(cos θ, −sin θ).
Thus, it has been found that the stream function of (4.67) represents
a uniform ﬂow of velocity U around a sphere of radius a (Fig. 4.6).
9
The formula (4.61) for Ψ=
1
2
Ur
2
sin
2
θ leads to (v
r
, v
θ
) = U(cos θ, −sin θ) = Ui.
10
Equation (4.60) has an inhomogeneous term F. However, because of the forces
associated with each Ψ
Us
described below, the linear combination is still valid.
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4.8. Low Reynolds number ﬂows 67
Fig. 4.6. Low Reynolds number ﬂow around a sphere: σ
θr
= 2µe
θr
, σ
rr
=
−p + 2µe
rr
.
Next, let us consider the force acting on the sphere of radius a
(i.e. the drag (D
i
)), which is given by the integral of stress over the
sphere surface S
a
:
D
i
=

S
r
σ
ij
n
j
dS, (n
r
, n
θ
, n
φ
) = (1, 0, 0), (4.70)
where σ
ij
is the stress tensor deﬁned by (4.4). In the present case,
σ
ij
takes the form,
σ
ij
= −pδ
ij
+ 2µe
ij
, (4.71)
where the Reynolds stress term ρ
0
v
i
v
j
is absent in the Stokes equa-
tion. The rate-of-strain tensors e
ij
in the spherical polar coordinates
are deﬁned by (D.28) and (D.29) in Appendix D.2. Using (4.68) and
(4.69) and setting r = a, we obtain the rate-of-strain tensors in the
frame (r, θ, φ) as follows:
e
θr
= −
3
4
U
a
sin θ, e
rr
= e
θθ
= e
φφ
= e
θφ
= e
φr
= 0. (4.72)
The pressure p of the Stokeslet is given by p
S
of (4.65), while the
pressure of the ﬁrst two terms of (4.67) is a constant p
0
, as shown in
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68 Viscous ﬂuids
Sec. 4.8.1, because they represent irrotational ﬂows. Thus, we have
p = −
3
4
Uap
S
+p
0
= −
3
2
µU
a
cos θ +p
0
. (4.73)
From (4.71)–(4.73), we can estimate the drag of the sphere by (4.70),
where dS = 2πa
2
sin θ dθ.
However, there is a more compact way to ﬁnd the drag on the
sphere. It can be shown that the irrotational ﬂows of the ﬁrst two
terms do not give rise to any force on the sphere (Problem 4.8).
Therefore, the force is due to the Stokeslet only, which is given by
F = −
3
4
Ua 8πµ = −6πµaU (toward the negative x direction). By
the law of reaction, the drag D experienced by the sphere toward the
positive x direction is given by
D = −F = 6πµaU. (4.74)
This is called the Stokes’ law of resistance for a moving sphere.
Stokes (1850) applied the formula (4.74) to a tiny cloud droplet
of radius a of a mass m falling with a steady velocity U in air
under the gravity force mg, and explained the suspension of clouds.
Then we have the equality,
6πµaU = mg.
Since the drop mass is proportional to a
3
, the falling velocity U is
proportional to a
2
and µ
−1
, becoming negligibly small as a →0.
4.9. Flows around a circular cylinder
When a long rod of circular cross-section is placed in a uniform
stream of water, it is often observed that vortices are formed on
the back of the body. The vortices sometime form a regular queue,
and sometime ﬂuctuate irregularly. On the other hand, if a rod is
moved with some speed with respect to water at rest, then similar
vortex formations are observed too. The hydrodynamic problem of
motion of a rod and associated generation of vortices are studied as
Flow of a viscous ﬂuid around a circular cylinder.
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4.10. Drag coeﬃcient and lift coeﬃcient 69
When a ﬂow has the same velocity vector U at all points in the
ﬂow without presence of a rod, the ﬂow is called a uniform ﬂow. If
a circular cylinder of diameter D is placed in such a uniform ﬂow, a
characteristic ﬂow pattern is observed downstream of the body. Such
a ﬂow pattern is called the wake. Particularly well-known wake is
the K´ arm´ an vortex street, observed at some values of the Reynolds
number R
e
= UD/ν (observed when 70 R
e
200 experimentally
and computationally). The K´arm´ an vortex street consists of two lines
of vortices with one line having a common sense of rotation but
the other line having its opposite and the vortices being located at
staggered positions. At much higher values of the Reynolds number,
the wake becomes irregular and turbulent, but there is still some
periodic component immersed in the turbulent wake.
However, far downstream from a body in the uniform ﬂow in the
x direction, the velocity decays and the stream lines become nearly
parallel (but not exactly parallel). There the ﬂow may be regarded as
a certain kind of boundary layer with its breadth being relatively thin
compared with the distance from the body. The pressure variation
across the wake may be very small. The limiting form of the wake is
considered in Problem 4.6.
Flow around a circular cylinder is the most typical example
of ﬂows around a body, and regarded as exhibiting the Reynolds
similarity in an idealistic way, where the Reynolds number R
e
plays
the role of control parameter exclusively. Various types of ﬂows are
summarized in Table 4.2. Figure 4.7 shows stream-lines of such ﬂows
obtained by computer simulations. Figure 1.2 is a photograph of an
air ﬂow around a circular cylinder at R
e
= 350.
4.10. Drag coeﬃcient and lift coeﬃcient
Suppose that a body is ﬁxed in a uniform stream of velocity U, and
subjected to a drag D D and a lift L. The drag is deﬁned as a force
on the body in the direction of the stream, while the lift is deﬁned
as a force on the body perpendicular to the stream direction.
In ﬂuid mechanics, these forces are normalized, and they are repre-
sented as dimensionless coeﬃcients of drag and lift. They are deﬁned
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70 Viscous ﬂuids
Table 4.2. A sequence of ﬂows around a circular cylinder (diameter
D) with respect to the Reynolds number R
e
= UD/ν with uniform
velocity U in the direction of x axis (horizontal), the z axis coinciding
with the cylinder axis, and the y axis taken vertically.
R
e
5 : Slow viscous ﬂows, with the stream-lines
symmetric w.r.t. x axis (up–down symmetry), and
symmetric w.r.t. y axis (fore–aft symmetry).
5 R
e
40 : A pair of steady separation eddies are formed on
the rear side of the cylinder.
The fore–aft symmetry is broken.
40 R
e
70 : Wake oscillations are observed: Hopf bifurcation.
The up–down symmetry is broken.
70 R
e
500 : K´arm´an vortex streets are observed in the wake.
Nonlinear modulation to the Hopf bifurcation.
500 R
e
: Wake is in turbulent state on which K´arm´an
vortex street is superimposed.
Transition to turbulent wake.
as follows.
Drag coeﬃcient : C
D
=
D
1
2
ρ U
2
S
, (4.75)
Lift coeﬃcient : C
L
=
L
1
2
ρU
2
S
, (4.76)
where S is a speciﬁc reference surface area (cross-section or others),
regarded as appropriate for the nomalization.
4.11. Problems
Problem 4.1 Hagen–Poiseuille ﬂow
(i) Using the cylindrical coordinates (x, r, θ) (Appendix D.2, by
rearranging from (r, θ, z) to (x, r, θ)), show that Eq. (4.39) can
be written as
1
r
∂
∂r

(4.79)
(Fig. 4.8). In addition, express a representative thickness δ of the
oscillating boundary layer in terms of ω and viscosity ν.
Problem 4.3 Taylor–Couette ﬂow
Suppose that there is a viscous ﬂuid between two concentric cylinders
of radius r
1
and r
2
(> r
1
) and the cylinders are rotating steadily
with angular velocity Ω
1
and Ω
2
, respectively (Fig. 4.9). Determine
the steady velocity distribution v(r), governed by (4.56).
Problem 4.4 Blasius ﬂow
(i) By applying the estimates of order of magnitude described in
Sec. 4.5, show the consistency of the approximate Eq. (4.32)
Fig. 4.8. Oscillating boundary layer.
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4.11. Problems 73
Fig. 4.9. Taylor–Couette ﬂow.
and describe what is the order of the neglected term νu
xx
. In
addition, show that Eq. (4.30) can be replaced by
∂p/∂y = 0. (4.80)
(ii) The velocity (u, v) in the boundary layer along a ﬂat plate is gov-
erned by Eq. (4.32) with p
x
= 0. By assuming that the stream
function takes the following similarity form,
ψ(x, y) =

νUx
1
2
f(η), η = y

U
νx
1
2
, (4.81)
under the boundary conditions (4.27) and (4.28), show that the
equation for f(η) and the boundary conditions are written as
f

→0 as ζ →±∞. (4.87)
(ii) Show that the following integral M is independent of x:
M = ρ

∞
−∞
u
2
dy,
where M is the momentum-ﬂux.
(iii) Determine the function f(ζ) and the jet velocity proﬁle u(x, y).
This is called the Bickley jet.
Problem 4.6 Wake ﬂow (2D)
We consider a two-dimensional steady ﬂow around a cylindrical body.
Suppose that the cylindrical body is ﬁxed in a uniform stream of
velocity U in the x direction and the (x, y)-plane is taken perpendic-
ularly to the cylindrical axis with the origin inside the cross-section.
In the wake far downstream from the body, the velocity decays
signiﬁcantly and the stream-lines are nearly parallel, and the pres-
sure variation across the wake would be very small (Fig. 4.12). The
velocity in the steady wake may be written as (U−u, v), where u > 0
is assumed, and Eq. (4.32) could be linearized with respect to small
components u and v, as follows:
U∂
x
u = ν ∂
2
y
u, (4.88)
with the boundary condition for u > 0,
u →0, as y →±∞. (4.89)
(i) Determine the velocity u(x, y) of the steady wake satisfying the
above Eq. (4.88) and the boundary condition. Furthemore, show
the following integral (volume ﬂux of inﬂow) is independent of x:

∞
−∞
u(x, y) dy = ρUQ. (4.90)
[Hint: The Momentum Theorem may be applied. Namely, the drag
is related to a momentum sink in the ﬂow, which can be estimated
by the momentum conservation integral (3.23) over a large volume
including the body. This result shows D ∝ ρQ (mass ﬂux of dragging
ﬂuid).]
Problem 4.7 Stokeslet
Derive the expression (4.66) of the force F of the Stokeslet (4.62),
according to the drag formula (4.70), by using (4.63) and (4.65).
Problem 4.8 Force by an irrotational ﬂow
Show that an irrotational ﬂow outside of a closed surface S does not
give rise to any force on S if there is no singularity out of S, according
to the drag formula (4.70) and Stokes equation (4.60).
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Chapter 5
Flows of ideal ﬂuids
Theory of ﬂows of ideal ﬂuids not only provides us the basis of study
of ﬂuid ﬂows, but also gives us fundamental physical ideas of con-
tinuous ﬁelds in Newtonian mechanics. Extending the ideas of ideal
ﬂuid ﬂows, the physical concepts can be applied to extensive areas
in physics and mathematics.
Governing equations of ﬂows of an ideal ﬂuid derived in Chapter 3
are summarized as follows:
∂
t
ρ + div(ρ v) = 0, (5.1)
∂
t
v + (v · ∇)v = −
1
ρ
grad p +f , (5.2)
∂
t
s + (v · ∇)s = 0. (5.3)
Boundary condition for an ideal ﬂuid ﬂow is that the normal com-
ponent of the velocity vanishes on a solid boundary surface at rest:
v
n
= v · n = 0, (5.4)
where n is the unit normal to the boundary surface. If the boundary
is in motion, the normal component v
n
of the ﬂuid velocity should
coincide with the normal velocity component V
n
of the boundary
v
n
= V
n
. (5.5)
Tangential components of both of the ﬂuid and moving boundary do
not necessarily coincide with each other in an ideal ﬂuid.
77
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78 Flows of ideal ﬂuids
5.1. Bernoulli’s equation
One of the basic theorems of ﬂows of an ideal ﬂuid is Bernoulli’s
theorem, which can be derived as follows. Suppose that the ﬂuid’s
entropy is uniform, i.e. the entropy s per unit mass of ﬂuid is constant
everywhere, and the external force has a potential χ represented by
f = −grad χ. Then, Eq. (5.2) reduces to (3.30), which is rewritten
here again:
∂
t
v +ω ×v = −grad

|v|
2
2
+h +χ

, (5.6)
where h is the enthalpy (h = e + p/ρ) and e the internal energy.
From this equation, one can derive two important equations. One is
the case of irrotational ﬂows for which ω = 0. The other is the case
of steady ﬂows. Here, we consider the latter case.
In steady ﬂows, the ﬁeld variables like the velocity is independent
of time t, and their time derivatives vanish identically. In this case,
we have
ω ×v = −grad

= 0, (5.8)
where v = |v|. The derivative ∂/∂s in the middle is the diﬀerentiation
along a stream-line parameterized with a variable s.
1
Thus, it is
1
According to the relation (1.9) along a stream-line, v · grad = (∂x/∂s)∂
x
+
(∂y/∂s)∂
y
+ (∂z/∂s)∂
z
= ∂/∂s.
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5.1. Bernoulli’s equation 79
z
y
stream-line
vortex-line
x
Bernou
ille
s
u
r
f
a
c
e
(
H
=
c
o
n
s
t
)
Fig. 5.1. Bernoulli surface.
found that
H :=
v
2
2
+h +χ = const. along a stream-line . (5.9)
This is called the Bernoulli theorem. The function H is constant
along a stream-line. Its value may be diﬀerent for diﬀerent stream-
lines.
In view of the property that the inner product of ω and ω × v
vanishes as well, the function H is constant along a vortex-line
2
(deﬁned in Sec. 5.3) analogously. This implies that the surface given
by H(x, y, z) = const. is covered with a family of stream-lines and
the vortex-lines crossing with the stream-lines. In most ﬂows, they
are not parallel.
3
The surface deﬁned by H = const. is called the
Bernoulli surface (Fig. 5.1).
The Bernoulli equation (5.9) is often applied to such a case where
the ﬂuid density is a constant everywhere, denoted by ρ
0
, and χ =
gz (g is the constant gravity acceleration). Then Eq. (5.9) can be
written as
4
ρ
0
H = ρ
0
v
2
2
+p +ρ
0
gz = const., (5.10)
2
A vortex-line is deﬁned by dx/ω
x
= dy/ω
y
= dz/ω
z
where ω = (ω
x
, ω
y
, ω
z
).
3
The ﬂow in which v is parallel to ω is said to a Beltrami ﬂow.
4
Daniel Bernoulli (1738) derived this Eq. (5.10), actually a form equivalent to it,
in the hydraulic problem of eﬄux. He coined the word Hydrodynamica for the
title of this dissertation.
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80 Flows of ideal ﬂuids
where h = p/ρ
0
+ e is used, and the internal energy e is included in
the “const.” on the right since e is constant in a ﬂuid of constant
density and constant entropy (see (3.27)).
When the gravity is negligible, the Bernoulli equation reduces to
1
2
ρ
0
v
2
+p = p
0
, p
0
: a constant. (5.11)
We consider an example of its application. Suppose that a two-
dimensional cylindrical body is immersed in a uniform ﬂow of velocity
U in the x-direction, and a steady ﬂow around the body is main-
tained in the (x, y)-plane. At distances far from the body, the veloc-
ity v = (u, v) tends to (U, 0), and the pressure tends to a constant
value p
∞
(Fig. 5.2). The Bernoulli equation (5.11) holds for each
stream-line. However at inﬁnity, the value on the left is given by the
same value
1
2
ρ
0
U
2
+p
∞
for all the stream-lines. Hence, the constant
on the right is the same for all the stream-lines. Thus, we have the
following equations valid for all the stream-lines:
1
2
ρ
0
v
2
+p =
1
2
ρ
0
U
2
+p
∞
. (5.12)
There exists always a dividing stream-line coming from the upstream
which divides the ﬂow into one going around the upper side B
+
of
the cylindrical body and the one going around its lower side B
−
(Fig. 5.2). The dividing stream-line terminates at a point on the body
surface, where the ﬂuid velocity necessarily vanishes. This point is
termed the stagnation point (see Problem 5.1), where the pressure p
s
A
B+
(U, 0) (U, 0)
O
B−
Fig. 5.2. Steady uniform ﬂow around a body.
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5.2. Kelvin’s circulation theorem 81
is given by
p
s
= p
∞
+
1
2
ρ
0
U
2
.
Namely, the stagnation pressure is higher than the pressure at inﬁn-
ity by the value
1
2
ρ
0
U
2
. This value
1
2
ρ
0
U
2
is called the dynamic pres-
sure, whereas the pressure p on the left of (5.12) is called the static
pressure.
5.2. Kelvin’s circulation theorem
A fundamental theorem of ﬂows for an ideal ﬂuid is the circulation
theorem. Given a velocity ﬁeld v(x, t) = (u, v, w), the circulation
along a closed curve C in a ﬂow ﬁeld is deﬁned by
Γ(C) :=

C
v · dl =

C
(udx +v dy +wdz), (5.13)
where

denotes an integral along a closed curve C, and dl =
(dx, dy, dz) denotes a line element along C.
An important property of ﬂuid motion can be derived if the closed
curve C is a material curve, i.e. each point on the curve moves with
the velocity v of ﬂuid (material) particle a. In this sense, the mate-
rial curve and material line element are denoted as C
a
and dl
a
,
respectively.
In the ﬂow governed by Eq. (5.6), the Kelvin’s circulation the-
orem (Kelvin, 1869) reads as follows. The circulation Γ
a
along the
material closed curve C
a
is invariant with respect to time (Fig. 5.3),
C
a
(t)
C
a
(0)
a
particle a
particle path
a
Fig. 5.3. Motion of a material closed curve.
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82 Flows of ideal ﬂuids
namely,
Γ
a
=

C
a
v · dl
a
is independent of t, (5.14)
or
D
Dt

C
a
v · dl
a
= 0. (5.15)
Since each line element dl
a
of C
a
moves with the ﬂuid material, the
time derivative must be the Lagrange derivative D/Dt. The theorem
is veriﬁed as follows.
Euler’s equation of motion (3.29) is rewritten as
Dv
Dt
= −grad(h +χ). (5.16)
In addition, it is noted that, denoting the velocities of two neighbor-
ing ﬂuid particles at x
a
and (x +δx)
a
by v and v +δv respectively,
we have the relation
D
Dt
δx
a
= δv, (5.17)
since D/Dt(x +δx)
a
= Dx
a
/Dt + Dδx
a
/Dt (see (1.32)).
Now, the time derivative DΓ
a
/Dt is given by
D
Dt

C
a
v · dl
a
= lim
¸
n
D
Dt

v
(n)
· dl
(n)
a

=

C
a
D
Dt
(v · dl
a
),
where the curve C
a
is divided into a number of small line elements
with n denoting its nth element and
¸
n
denotes the summation with
respect to all the elements. “lim” denotes taking a limit in such a way
that the total number N of elements is made inﬁnite by keeping each
line element inﬁnitesimally small. Thus we have
D
Dt
Γ
a
=

C
a
d(h +χ) = 0,
since dx · grad f = df for a scalar function f in general. This is due
to single-valuedness of h and χ. Thus Eq. (5.15) has been veriﬁed.
In summary, the circulation theorem has been veriﬁed under the
condition of Euler equation (5.16) together with the homentropy of
the ﬂuid and the conservative external force, but the ﬂuid density is
not necessarily assumed constant.
The same circulation theorem can be proved as well for the case
of a uniform density (ρ is constant), or the case of a barotropic ﬂuid
(in which the pressure p is a function of density ρ only, i.e. p = p(ρ)),
instead of the homentropic property of uniform value of entropy s.
5.3. Flux of vortex-lines
Given a velocity ﬁeld v(x), the vorticity ω = (ω
x
, ω
y
, ω
z
) is deﬁned
by ∇ × v. In an analogous way to the stream-line, a vortex-line is
deﬁned by
dx
ω
x
(x, y, z)
=
dy
ω
y
(x, y, z)
=
dz
ω
z
(x, y, z)
=
dl
|ω|
, (5.19)
where the vector ω is tangent to the vortex-line and l is the arc-length
along it (Fig. 5.4).
The circulation Γ of (5.13) is also represented with a surface inte-
gral in terms of ω by using the Stokes theorem (A.35) as
Γ[C] =

C
v · dl =

S
(∇×v) · ndS =

S
ω · ndS, (5.20)
where S is an open surface immersed in a ﬂuid and bounded by the
closed curve C with n being a unit normal to the surface element
dS (Fig. 5.5). The integrand of the surface integral on the right
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84 Flows of ideal ﬂuids
Fig. 5.4. Vortex-line and tube.
Fig. 5.5. Closed curve C and surface S.
represents the magnitude of ﬂux of vortex-lines passing through the
surface element:
ω
n
dS. (5.21)
Hence, the circulation Γ[C] is equal to the total magnitude of ﬂux of
vortex-lines passing through the closed curve C.
Thus, the Kelvin’s circulation theorem is rephrased in the follow-
ing way. The total magnitude of ﬂux of vortex-lines passing through
a material closed curve C
a
, i.e. circulation Γ[C
a
], is invariant with
respect to the time for the motion of an ideal ﬂuid of homentropy (or
uniform density, or barotropy) under the conservative external force.
A vortex tube is deﬁned as a tube formed by the set of all vortex-
lines passing through the closed curve C. The strength k, i.e. circu-
lation, of a small vortex tube of an inﬁnitesimal cross-section δS is
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5.4. Potential ﬂows 85
given by k = ω
n
δS, which is invariant along the tube itself. This is
veriﬁed by using the property (a vector identity (A.26)),
div ω = ∇· (∇×v) = 0, (5.22)
because there is no ﬂux out of the tube surface.
5.4. Potential ﬂows
From the Kelvin circulation theorem, one can deduce an important
property. Suppose that the ﬂow ﬁeld is steady, hence stream-lines
are unchanged. In addition, suppose that the vorticity curl v is zero
in a domain D including a point P. Consider a small closed curve
C within D encircling a stream-line L passing through the point P.
Then the circulation Γ[C] vanishes by (5.20). The circulation theorem
assures that the zero-value of the circulation Γ[C] is kept invariant
during the translational motion of the closed curve C along the line
L carried by the steady ﬂuid ﬂow. Namely, once the vorticity was
zero in a domain D

including a point on a stream-line L, then the
vorticity is zero (curl v = 0) in an extended domain D

including
the stream-line L which is swept by D

carried along by the ﬂow. If
the velocity ﬁeld is unsteady, the property curl v = 0 is maintained
in a domain carried along by the moving ﬂuid.
The ﬁeld where the vorticity is zero (curl v = 0) is termed irrota-
tional. Any vector ﬁeld v(x) whose curl vanishes in a simply-connected
domain D is represented in terms of a potential function Φ(x) as
v = grad Φ = (∂
x
Φ, ∂
y
Φ, ∂
z
Φ) = ∇Φ (5.23)
(see Appendix B.1). It is obvious by the vector identity (A.25) that
this is suﬃcient.
5
In this sense, an irrotational ﬂow is also called a
potential ﬂow. The scalar function Φ(x) is called the velocity poten-
tial. The ﬂow in which the vorticity is not zero is said to be rotational.
Suppose that a body is immersed in an otherwise uniform ﬂow and
that the newly formed ﬂow is steady. At upstream points of inﬁnity,
5
It can be veriﬁed that this is also necessary, as veriﬁed in Appendix B.1.
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86 Flows of ideal ﬂuids
the vorticty is zero because the velocity ﬁeld is uniform there. Thus,
we ﬁnd that this is a potential ﬂow.
For a potential ﬂow, setting ω = 0 in the equation of motion (5.6),
we have
∂
t
v + grad

|v|
2
2
+h +χ

= 0. (5.24)
We can integrate this Euler’s equation of motion. Substituting the
expression (5.23) for v and writing as v = |v|, we obtain
grad

∂Φ
∂t
+
v
2
2
+h +χ

= 0. (5.25)
This means that the expression in ( ) is a function of time t only,
∂
t
Φ +
1
2
v
2
+h +χ = f(t), (5.26)
where f(t) is an arbitrary function of time. However, if we introduce
a function Φ

by Φ

= Φ −

t
f(t

)dt

, we obtain
grad Φ

= grad Φ = v, ∂
t
Φ

= ∂
t
Φ −f(t).
Then, Eq. (5.26) reduces to
∂
t
Φ

+
1
2
v
2
+h +χ = const.
This (or (5.26)) is called the integral of the motion, where
v
2
= (∂
x
Φ)
2
+ (∂
x
Φ)
2
+ (∂
x
Φ)
2
. (5.27)
Henceforce, in the above integral, the function Φ is used instead
of Φ
6
:
∂
t
Φ +
1
2
v
2
+h +χ = const. (5.28)
When the ﬂow is steady, the ﬁrst term ∂
t
Φ vanishes. Then we
obtain the same expression as Bernoulli’s equation (5.9). However
6
In this case, the function Φ is to be supplemented with an arbitrary function of
time. The invariance with the transformation Φ → Φ

may be termed as gauge
invariance, which is analogous to that of electromagnetism. See Chapter 12.
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5.5. Irrotational incompressible ﬂows (3D) 87
in the present potential ﬂow, the constant on the right holds at all
points (not restricted to a single stream-line).
If the gravitational potential is written as χ = gz, the integral of
motion (5.28) for a ﬂuid of uniform density ρ
0
is given as
∂
t
Φ +
1
2
v
2
+ (p/ρ
0
) +gz = const. (5.29)
When the velocity potential Φ is known, this equation gives the pres-
sure p since v
2
is also known from (5.27).
5.5. Irrotational incompressible ﬂows (3D)
7
An irrotational ﬂow has a velocity potential Φ, and the velocity is
represented as v = grad Φ. In addition, if the ﬂuid is incompressible,
i.e. div v = 0, then we have the following equation,
div v = ∇· ∇Φ = ∇
2
Φ = 0, (5.30)
where ∇
2
= ∂
2
x
+∂
2
y
+∂
2
z
. This is also written as
0 = ∇
2
Φ = Φ
xx
+ Φ
yy
+ Φ
zz
= ∆Φ. (5.31)
This is the Laplace equation for Φ where ∆ is the Laplacian. In gen-
eral, the functions Φ(x, y, z) satisfying the Laplace equation is called
harmonic functions. Thus, it is found that irrotational incompressible
ﬂows are described by harmonic functions.
In other words, any harmonic function Φ (satisfying ∆Φ = 0)
represents a certain irrotational incompressible ﬂow. To see what
kind of ﬂow is represented by Φ, we have to examine what kind of
boundary conditions are satisﬁed by the function Φ.
When a body is ﬁxed in an irrotational ﬂow, the normal compo-
nent v
n
of the velocity must vanish on the surface S
b
of the body:
v
n
= n · grad Φ = ∂Φ/∂n = 0, (5.32)
where n is a unit normal to S
b
.
7
3D: three-dimensional, 2D: two-dimensional.
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88 Flows of ideal ﬂuids
It is remarkable that Eq. (5.31) is linear with respect to the veloc-
ity potential Φ (as well as the boundary condition (5.32) is so). If
two functions Φ
1
and Φ
2
satisfy (5.31), then their linear combination
Φ
1
+ Φ
2
does satisfy it too. Often the combined potential Φ
1
+ Φ
2
satisﬁes a required boundary condition even if each potential does
not. Then we get a required solution Φ
1
+ Φ
2
.
A Neumann problem for harmonic functions is a boundary value
problem in the theory of partial diﬀerential equations, which is to
ﬁnd a function Φ satisfying the Laplace equation ∆Φ = 0 together
with the boundary condition: ∂Φ/∂n given on a surface S. In the
context of ﬂuid dynamics, if ∂Φ/∂n = 0 on the boundary S and
grad Φ →U at inﬁnity (with U a constant vector), this is a problem
to determine the velocity potential Φ of a ﬂow around a solid body
S at rest in a uniform stream U.
5.6. Examples of irrotational incompressible
ﬂows (3D)
A simplest example is given by a linear function, Φ = Ax+By +Cz
(with A, B, C: constants), which satisﬁes the Laplace equation ∆Φ =
0 identically. The velocity v = grad Φ = (A, B, C) is a constant
vector at every point, and hence this expresses a uniform ﬂow.
In particular, Φ
U
= Ux is the velocity potential of a uniform ﬂow
of velocity U in the x direction.
5.6.1. Source (or sink)
Consider a point x = (x, y, z) in the cartesian (x, y, z) coordinates.
In the spherical polar coordinates (r, θ, φ) (see Appendix D.3), the
radial coordinate r denotes its distance from the origin. A ﬂow due
to a source (or a sink) located at the origin is represented by the
velocity potential Φ
s
:
Φ
s
= −
m
r
, (5.33)
where r =

= −
1
r
2
∂
∂r
(−m) = 0, for r = 0,
where a singular point r = 0 (the origin) is excluded. This veriﬁes
that the function Φ
s
is a harmonic function except when r = 0. The
property that the function Φ
s
is the velocity potential describing
a ﬂow due to a source can be shown by its velocity ﬁeld, which is
given by
v = grad

−
m
r

=
m
r
2

x
r
,
y
r
,
z
r

,
since ∂r/∂x = x/r, etc. Taking a sphere S
R
of an arbitrary radius R
with its center at the origin, the volume ﬂux of ﬂuid ﬂow out of S
R
per unit time is given by

V
R
∇
2
Φ
s
dV = 4πm,
where V
R
is the spherical volume of radius R with its center at the
origin. This implies

∇
2
(1/r)dV = −4π, namely
∇
2
1
r
= −4πδ(x) δ(y) δ(z), (5.34)
(see Appendix Sec. A.7 for the deﬁnition of δ-function).
8
Thus, it is
found that the ﬂux out of the spherical surface is independent of its
radius and equal to a constant 4πm. Therefore, the velocity potential
Φ
s
= −m/r describes a source of ﬂuid ﬂow from the origin if m > 0,
or a sink if m < 0. Figure 5.6 shows such a source ﬂow (left), and a
sink ﬂow (right) in the upper half space.
8
The function 1/(4πr) is called a fundamental solution of the Laplace equation
due to the property (5.34).
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90 Flows of ideal ﬂuids
Fig. 5.6. (Left) source (m > 0); (right) sink for a half space (m < 0).
5.6.2. A source in a uniform ﬂow
One can consider a linear combination of two potentials:
Φ
Us
:= Φ
U
+ Φ
s
= Ux −
m
r
. (5.35)
What kind of ﬂow is represented by Φ
Us
? The velocity ﬁeld is given by
v = (U, 0, 0) +
m
r
2

x
r
,
y
r
,
z
r

,
in the cartesian (x, y, z) space. At large distances as r → ∞, the
velocity v approaches the ﬁrst uniform ﬂow. On the x axis (y = 0,
z = 0), the x component velocity is given by u = U+mx/r
3
, whereas
the other two components are zero. There is a stagnation point where
the velocity vanishes. The x component u vanishes at a point x = −r
s
(r
s
> 0, y = 0, z = 0) on the negative x axis, where r
s
is given by
r
s
=

m/U.
Evidently the ﬂow ﬁeld has a rotational symmetry with respect to
the x axis. The stream-lines emanating from the stagnation point
forms a surface of rotation symmetry S
∗
around the x axis, which
divides the ﬂuid coming out of the origin (the source) from the ﬂuid
of the on-coming uniform ﬂow from the left (Fig. 5.7). At suﬃciently
downstream, this surface S
∗
tends to the surface of a circular cylinder
of radius a (say). The rate of outﬂow per unit time is 4πm as given
in the previous case. This amount is equal to the uniform ﬂow of
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5.6. Examples of irrotational incompressible ﬂows (3D) 91
Fig. 5.7. A source at O in a uniform ﬂow of velocity U. The external ﬂow is
equivalent to a ﬂow around a semi-inﬁnite cylindrical body B
∗
.
velocity U passing through the circular cross-section of radius a.
Thus, we have
4πm = πa
2
U, ∴ a = 2

m/U
One can replace the surface S
∗
(and its inside space) by a solid
body B
∗
, which is a semi-inﬁnite cylinder with a rounded head. The
ﬂow around the solid body B
∗
is given by the same velocity potential
of (5.35) (if the body axis coincides with the uniform ﬂow) with the
restriction that only the ﬂow external to B
∗
is considered.
5.6.3. Dipole
Diﬀerentiating the source potential Φ
s
of (5.33) with respect to x, we
obtain ∂
x
Φ
s
= mx/r
3
. This form x/r
3
is called a dipole potential.
The reason is as follows.
Suppose that there is a source −mf(x, y, z) at the origin (0, 0, 0)
and a sink mf(x + , y, z) at (−, 0, 0), where f(x, y, z) is deﬁned
by 1/

cos θ, (5.38)
since x = r cos θ. This is an axisymmetric ﬂow. In fact, using (A.30),
or (D.17) for the ∇ operator of the spherical coordinates (r, θ, φ) in
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5.6. Examples of irrotational incompressible ﬂows (3D) 93
Appendix D.3, the velocity in the spherical frame is given by
v =

U

1 −
2µ
Ur
3

cos θ, −U

1 +
µ
Ur
2

sin θ, 0

. (5.39)
At inﬁnity as r →∞, the velocity tends to the uniform ﬂow (U cos θ,
−U sinθ, 0) in the spherical frame, which is equivalent to (U, 0, 0) in
the (x, y, z) frame.
The radial component v
r
is given by v
r
= U(1 −(2µ/Ur
3
)) cos θ.
From this, it is found that there is a sphere of radius a = (2µ/U)
1/3
on
which the normal velocity component v
r
vanishes for all θ (Fig. 5.9).
This implies that the expression (5.39) represents the velocity of a
ﬂow around a sphere of radius a. A dipole placed in a uniform ﬂow
can represent a ﬂow ﬁeld around a sphere placed in a uniform stream.
If the radius a is given in advance, the dipole strength µ should be
1
2
Ua
3
.
Thus, it is found that the velocity potential of the uniform ﬂow
of velocity U around a sphere of radius a is given by
Φ
Usph
= Ux +
1
2
Ua
3
x
r
3
= U

, (5.57)
where the overline denotes complex conjugate.
The deﬁnition of the complex function F(z) = Φ + iΨ by (5.53)
is regarded as a map from a complex variable z = x +iy to another
9
The form d(Φ +iΨ) = (u −iv)(dx +idy) was ﬁrst given by d’Alembert (1761).
The second expression of (5.55) is the derivative with ∆z = ∆x, whereas the third
one is that with ∆z = i∆y. Namely, diﬀerentiability means that the derivative
does not depend on the direction of diﬀerentiation.
10
¯
A ≡ a −ib is the complex conjugate of A = a +ib (a, b: real).
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98 Flows of ideal ﬂuids
complex variable F = Φ + iΨ, or an inverse map from F to z. This
is called the conformal map. In fact, in the (x, y)-plane, we have two
families of curves deﬁned by
Φ(x, y) = const., Ψ(x, y) = const. (5.58)
The second describes a family of stream-lines as explained in
Appendix B.2, while the ﬁrst may be called a family of equi-potential
curves. The two families of curves form an orthogonal net in the (x, y)-
plane (see Problem 5.3). In the (Φ, Ψ)-plane, the family of curves
Φ = const. are straight lines parallel to the imaginary axis, while the
family of curves Ψ = const. are parallel to the real axis. Thus the
two sets of orthogonal parallel straight lines in the (Φ, Ψ) plane are
mapped to the orthogonal net in the (x, y)-plane by the complex func-
tion F(z). The property of angle-preservation by a complex analytic
function F(z) is valid for any intersecting angle (see Problem 5.4).
From (5.55), we have
dF = wdz = (u −iv)(dx +idy) = (udx +vdy) +i(udy −vdx).
Integrating this along a closed curve C, we obtain

C
wdz =

C
(udx +vdy) +i

C
(udy −vdx) = Γ(C) +iQ(C).
(5.59)
The ﬁrst real part is just the velocity circulation Γ(C) along C,
according to (5.13). The second imaginary part, if multiplied by ρ,
is the mass ﬂux across C. In fact, we can write
Q(C) =

= [2πim] = 2πm. (5.64)
If C does not enclose the origin, we have Q(C) = 0. The circulation
Γ(C) along C is zero since the contour integral

C
wdz = 2πim is
purely imaginary with a real m.
Thus it is found that the complex potential F
s
describes a source
of ﬂuid ﬂow from the origin if m > 0, or a sink if m < 0.
5.8.2. A source in a uniform ﬂow
Consider the following complex potential:
F
Us
:= Uz +F
s
= Uz +mlog z. (5.65)
The complex velocity is given by
w =
dF
Us
dz
= U +
m
z
.
At large distances as |z| → ∞, the velocity w approaches the ﬁrst
uniform ﬂow w = U. There is a stagnation-point on the real axis (x
axis) where the velocity w vanishes (see Problem 5.2 for a stagnation-
point ﬂow). On the real axis, we have z = x, and the x component is
given by u = U+m/x, whereas the y component vanishes. u vanishes
at a point x
s
on the negative x axis, where x
s
= −m/U. The ﬂow
ﬁeld has a symmetry with respect to the x axis. The stream-lines
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5.8. Examples of 2D ﬂows represented by complex potentials 101
emanating from the stagnation point forms a curve C
∗
of mirror-
symmetry with respect to the x axis, which divides the ﬂuid emerging
out of the source at z = 0 from the ﬂuid of oncoming uniform ﬂow
from the left. At suﬃciently downstream, this curve C
∗
tends to two
parallel lines of distance d (say). The rate of outﬂow per unit time is
2πm as given in the previous example. This amount is equal to the
uniform ﬂow of velocity U between the two parallel lines of width d.
Thus, we have
2πm = Ud, ∴ d = 2πm/U
One can replace the curve C
∗
(and its inside space) by a solid body
C
∗
, which is a semi-inﬁnite plate of thickness d with a rounded
edge.
5.8.3. Dipole
A derivative of the source potential F
s
of (5.62) with respect to x
becomes a dipole potential. The reason is as follows.
Suppose that there is a source mlog z at the origin (0, 0) and a
sink −mlog(z +) at (−, 0), where m, > 0. The complex potential
of the combined ﬂow is given by
F = mlog z −mlog(z +)
= −m[log(z +) −log z]
= −m
∂
∂z
log z +O(
2
) = −m
1
z
+O(
2
).
Furthermore, consider the limit that the pair of source and sink
becomes closer indeﬁnitely as → 0, but that the product m
tends to a nonzero constant µ. Then, we obtain the following dipole
potential,
F
d
= −
µ
z
. (5.66)
where µ is termed as the strength of dipole in the x-direction. The
dipole in the y-direction is obtained by replacing by i, and hence
replacing µ by iµ.
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102 Flows of ideal ﬂuids
The function F
d
= −µ(d/dz) log z is an analytic function except
at the origin because the derivative of an analytic function log z
(z = 0) is another analytic function. According to the theory of
complex functions, if f(z) is an analytic function in a domain, then
the derivative d
n
f/dz
n
is another analytic function for any integer
n in the same domain.
5.8.4. A circular cylinder in a uniform ﬂow
The dipole −F
d
in a uniform stream Uz,
F
Ud
:= Uz −F
d
= Uz +
µ
z
(5.67)
describes a uniform ﬂow around a circle.
The complex velocity is given by dF
Ud
/dz, which is
w = u −iv = U −
µ
z
2
=

µ/U
on which the normal velocity component v
r
vanishes for all θ. This
implies that the expression (5.68) represents the velocity of a ﬂow
around a circle of radius a. A dipole (in the negative x-direction)
placed in a uniform ﬂow (in the positive x-direction) can represent a
ﬂow ﬁeld around a circle placed in a uniform stream. If the radius a
is given in advance, the dipole strength µ is given by Ua
2
.
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5.8. Examples of 2D ﬂows represented by complex potentials 103
Thus, it is found that the complex potential of the uniform ﬂow
of velocity U around a circle of radius a is given by
F
Ucirc
= U

C
k
2πi
dz
z
= 2πi
k
2πi
= k. (5.74)
Therefore, the velocity circulation round C enclosing z = 0 is given
by Γ(C) = k, whereas Q(C) = 0. If C does not enclose the origin,
we have Γ(C) = 0 and Q(C) = 0.
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104 Flows of ideal ﬂuids
Thus it is found that the complex potential F
v
= (k/2πi) log z
represents a concentrated vortex located at z = 0 with the circulation
k, counter-clockwise if k > 0, or clockwise if k < 0.
5.9. Induced mass
5.9.1. Kinetic energy induced by a moving body
Let us consider an ingenious analysis of potential ﬂow induced by
a solid body moving through an invisid ﬂuid otherwise at rest in
the three-dimensional space (x, y, z). This is a formulation in which
the total momentum and energy of ﬂuid motion thus induced can
be represented in terms of a constant tensor depending only on
the asymptotic behavior of velocity potential Φ(r) at inﬁnity, where
r = (x, y, z). [LL87, Sec. 11]
Potential ﬂow of an incompressible ﬂuid satisﬁes the Laplace equa-
tion ∇
2
Φ = 0. We investigate the ﬂuid velocity at great distances
from the moving body. Since the ﬂuid is at rest at inﬁnity, the veloc-
ity grad Φ must vanish at inﬁnity. We take the origin inside the mov-
ing body at a particular instant of time. We already know (Sec. 5.6)
that the function 1/r is a particular solution of the Laplace equation
(r = 0), where r is the distance from the origin. In addition, its space
derivatives of any order are also solutions. Their linear combination
is also a solution. All these solutions vanish at inﬁnity. Thus, the
general form of solutions of Laplace equation is represented by
Φ =
a
r
+A· grad
1
r
+ [higher order derivatives of 1/r], (5.75)
at large distances from the body, where a and A are constants inde-
pendent of coordinates. The constant a must be zero in incompress-
ible ﬂows since the ﬁrst term represents a ﬂow due to a ﬂuid source
at the origin (Sec. 5.6.1). The second term represents a ﬂow due to a
dipole at the origin (Sec. 5.6.3). The series form (5.75) is called the
multipole expansion.
We concentrate on the dipole term including A, since the terms
of higher order could be neglected at large distances in the following
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5.9. Induced mass 105
analysis. Then, we have
Φ = A· grad
1
r
= −
A· e
r
2
, e ≡
r
|r|
, (5.76)
where r = (x, y, z) is the radial vector. The velocity is given by
v = grad Φ = (A· grad) grad
1
r
=
3(A· e)e −A
r
3
. (5.77)
It is remarkable that the velocity diminishes as r
−3
at large distances,
which is a characteristic property of dipoles. The constant vector A
depends on the detailed shape of the body.
12
The vector A appearing in (5.76) can be related to total momen-
tum and energy of ﬂuid motion induced by the moving body. The
total kinetic energy is
13
E =
1
2
ρ

V
∗
|v|
2
dV,
where the integration is to be taken over all space outside the body,
denoted by V
∗
. The integral in the unbounded space is dealt with
in the following way. First, we choose a region of space V bounded
by a sphere S
R
of a large radius R with its center at the origin and
integrate over V . Next, we let R tend to inﬁnity. We denote the
volume of the body by V
b
and its surface by S
b
.
Denoting the velocity of the body by U, we note the following
identity:

V
∗
|v|
2
dV =

V
∗
|U|
2
dV +

V
∗
(v −U) · (v +U) dV.
The ﬁrst integral on the right is simply U
2
(V −V
b
), since U ≡ |U| is a
constant. In the second integral, we use the expressions: v = grad Φ,
12
The vector A can be determined by solving the equation ∇
2
Φ = 0 in the
complete domain, taking into account the boundary condition at the surface of
the moving body.
13
The internal energy of an incompressible ideal ﬂuid is constant.
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106 Flows of ideal ﬂuids
and U = grad(U· r) for the second factor. Then, we have
(v−U)· (v+U) = (v−U)· grad(Φ+U· r) = div[(Φ+U· r) (v−U)],
since div v = 0 and div U = 0. The second integral is now trans-
formed into a surface integral over S
R
and S
b
. Thus, we have

V
∗
|v|
2
dV = U
2
(V −V
b
) +

S
R
+S
b
(Φ +U· r) (v −U) · ndS,
where n is the unit outward normal to the bounding surface. On the
body’s surface S
b
, the normal component v·n must be equal to U· n.
Hence the surface integral over S
b
vanishes identically. On the remote
surface S
R
, we can use (5.76) for Φ and (5.77) for v. Then, we obtain

|v|
2
dV = U
2

4
3
πR
3
−V
b

+

3(A· e)(U· e)
−R
3
(U· e)
2
−2R
−3
(A· e)
2

dΩ, (5.78)
where, on the spherical surface S
R
, we used the expressions: n = e
and dS = R
2
dΩ with dΩ an element of solid angle (dΩ = sin θ dθdφ).
Here, we apply the following formula for the integral of (A · e)×
(B · e) over dΩ divided by 4π (equivalent to an average over a unit
sphere):
1
4π

(A· e)(B· e) dΩ =
1
3
A· B, (5.79)
for constant vectors A and B (see Problem 5.9 for its proof).
Carrying out the integration of (5.78) by using (5.79), neglecting
the last term (∝ R
−3
) since it vanishes as R → ∞ and dropping
the cancelling terms of (4/3)πR
3
U
2
, we ﬁnally obtain the following
expression for the total energy of ﬂuid motion (multiplying by
1
2
ρ):
E =
1
2
ρ

|v|
2
dV =
1
2
ρ

4π A· U−V
b
U
2

. (5.80)
In order to obtain the exact expression of A, we need a complete
solution of the Neumann problem of the Laplace equation,
∇
2
Φ = 0, for r ∈ V
∗
; n · ∇Φ = n · U on S
b
. (5.81)
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5.9. Induced mass 107
From this, we can make a deductive reasoning about the general
nature of the dependence of A on velocity U of the body. In view
of the properties that the governing equation is linear with respect
to Φ, and that the boundary condition for the normal derivative of
Φ is linear in U, it follows that the complete solution can be repre-
sented by a linear combination of potentials in the form of multipole
expansion (5.75), and that the coeﬃcient A of the dipole term must
be a linear function of the components of U, i.e. A
i
= c
ij
U
j
.
14
5.9.2. Induced mass
Thus, the energy E of (5.80) is represented by a quadratic function
of U
i
, and can be written in the following form:
E =
1
2
m
ij
U
i
U
j
, (5.82)
m
ij
= ρ

4πc
ij
−V
b
δ
ij

, (5.83)
where m
ij
is a certain symmetrical tensor (the expression (5.82)
enables symmetrization of m
ij
even if the original m
ij
is not so).
m
ij
is called the induced-mass tensor, or the added mass. The latter
meaning will become clear by the reasoning given next.
Knowing the energy E, we can obtain an expression of the total
momentum P of ﬂuid motion. Suppose that the body is subject to an
external force F. The momentum of ﬂuid will thereby be increased
by dP (say) during a short time dt. This increase is related to the
force by dP = Fdt. Scalar multiplication with the velocity U leads
to U· dP = F · Udt, i.e. the work done by the force F through the
distance Udt. This must be equal to the increase of energy dE of
14
The dipole potential Φ of (5.76) are of the form A
i
φ
i
by using three dipole
components φ
i
. On the other hand, with Φ = U
j
ψ
j
, the Neumann problem (5.81)
can be transformed to ∇
2
ψ
j
= 0 with the boundary condition n· ∇ψ
j
= n
j
. The
three solutions ψ
j
thus determined depend on the body shape, but are indepen-
dent of U
i
. ψ
j
may include a dipole term ψ
(d)
j
= c
ij
φ
i
in the asymptotic multipole
expansion at large distances. Therefore, we have A
i
φ
i
= U
j
ψ
(d)
j
= U
j
c
ij
φ
i
. From
this, we obtain A
i
= c
ij
U
j
. For the problem in Sec. 5.6.4 of a sphere of radius a
moving with (−U, 0, 0), we found A = (A
x
, A
y
, A
z
) =
`
1
2
a
3
(−U), 0, 0
´
.
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108 Flows of ideal ﬂuids
the ﬂuid. We already have an expression of E from (5.82). Hence, we
obtain two expressions of dE:
dE = U· dP, dE = m
ij
U
j
dU
i
,
where symmetry of the mass tensor m
ij
is used. From the equivalence
of the two expressions, we ﬁnd that
P
i
= m
ij
U
j
, or P = 4πρA−ρV
b
U. (5.84)
It is remarkable that the total ﬂuid momentumis given by a ﬁnite quan-
tity, although the ﬂuid velocity is distributed over unbounded space.
5.9.3. d’Alembert’s paradox and virtual mass
The momentum transmitted to the ﬂuid by the body is dP/dt per
unit time, the reaction force F

from the ﬂuid on the body is given by
F

= −dP/dt.
Suppose that the body is moving steadily in an ideal ﬂuid at a veloc-
ity U and inducing an irrotational ﬂow around it. Then, we should
have P = const., since U is constant, so we obtain F

= 0. Hence,
there would be no force. This is the result known as the d’Alembert
paradox. This paradox is clearly seen by considering the drag. The
presence of a drag in uniform motion of a body would mean that
work is continually done on the ﬂuid by the external force (required
to maintain the steady motion) and that the ﬂuid will gain energy
continually. However, there is no dissipation of energy in an ideal ﬂuid
by deﬁnition, and the velocity ﬁeld decays so rapidly with increasing
distance from the body that there can be no energy ﬂow to inﬁnity
(out of the space). Thus, no force is possible for uniform motion of a
body in an ideal ﬂuid.
15
If there was a nonzero force in such a case,
that would be a real paradox.
Suppose that a body of mass m is moving with acceleration under
the action of an external force f . This force must be equated to
15
Except for the lift L in the two-dimensional problem of uniform ﬂow around a
cylindrical body with a circulation (Problem 5.8). We have L · U = 0.
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5.10. Problems 109
the time derivative of the total momentum of the system. The total
momentum is the sum of the momentum mU of the body and the
momentum P of the ﬂuid. Thus,
m
dU
dt
+
dP
dt
= f . (5.85)
This can also be written as
(mδ
ij
+m
ij
)
dU
j
dt
= f
i
. (5.86)
This is the equation of motion of a body immersed in an ideal ﬂuid.
This clearly shows that m
ij
is an added mass (tensor). The factor
(mδ
ij
+ m
ij
) in front of the acceleration dU
j
/dt is the virtual mass
(tensor).
The added mass can be estimated from Φ
U sph
of Sec. 5.6.4 (and
from the remark of the footnote in Sec. 5.9.1), which implied c
ij
=
1
2
a
3
δ
ij
for the motion of a sphere of radius a. Hence, the formula
(5.83) with V
b
= (4π/3)a
3
yields
m
ij
=
1
2

0
= 0
O
Fig. 5.11. Stagnation-point ﬂow (Ψ: stream function, Φ: velocity potential).
Problem 5.3 Orthogonal net of a complex analytic
function
The expressions (5.58) deﬁne two families of equi-potential lines
and stream-lines. Show mutual orthogonal-intersection of the two
families.
Problem 5.4 Conformal property
Suppose that we are given two complex planes z = x + iy and
Z = X+iY , and that there is a point z
0
in the z plane and two points
z
1
and z
2
inﬁnitesimally close to z
0
. The two planes are related by a
complex analytic function Z = F(z), and the three points z
0
, z
1
, z
2
are mapped to Z
0
, Z
1
, Z
2
in the Z plane by F(z), respectively. Show
that the intersecting angle θ between two inﬁnitesimal segments
(from z
1
− z
0
to z
2
− z
0
) is invariant by the map F(z), namely the
intersecting angle between two inﬁnitesimal segments (from Z
1
−Z
0
to Z
2
−Z
0
) is θ.
Problem 5.5 Joukowski transformation
Suppose that we have two complex planes z = x+iy and ζ = ξ +iη,
and that we are given the following transformation between z and ζ:
z = f(ζ), f(ζ) = ζ +
a
2
ζ
(a : real positive), (5.88)
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5.10. Problems 111
called the Joukowski transformation.
(i) Using the polar representation ζ = σe
iφ
, show that the circle
σ = a in the ζ plane is mapped to a segment L on the real
axis of z plane: x ∈ [−2a, +2a], y = 0. In addition, show that
the exterior of the circle σ = a of the ζ plane is mapped to the
whole z plane except L (a cut along the real axis x), and that
the interior of the circle σ = a is mapped to the entire z plane
except L, too.
(ii) Show that the following potential F
α
(ζ) describes an inclined
uniform ﬂow around a circle of radius a in the ζ plane:
F
α
(ζ) = U

ζe
−iα
+
a
2
e
iα
ζ

(α : real). (5.89)
In addition, determine what is the angle of inclination with
respect to the real axis ξ [Fig. 5.12(a)].
(iii) Show that the corresponding ﬂow in the z plane is an inclined
uniform ﬂow around a ﬂat plate of length 4a [Fig. 5.12(b)].
Fig. 5.12.
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112 Flows of ideal ﬂuids
(iv) Show that the potential F
⊥
(ζ) represents a ﬂow impinging on a
vertical ﬂat plate (on the imaginary axis) in the Z plane at right
angles from left (from the negative real axis of Z) [Fig. 5.12(c)]:
F
⊥
(ζ) = −iU

ζ −
a
2
ζ

, Z = −i

ζ +
a
2
ζ

. (5.90)
Problem 5.6 Residue
Show that the following contour integral of a complex function F(z)
along C around z = 0:

C
F(z) dz = 2πi A
−1
,
F(z) =
∞
¸
n=−∞
A
n
z
n
= · · · +
A
−1
z
+ · · · ,
(5.91)
where C is a counter-clockwise closed contour around the origin z = 0.
The coeﬃcient A
−1
is called the residue at the simple pole z = 0.
Problem 5.7 Blasius formula
Suppose that a body B is ﬁxed within a two-dimensional incompress-
ible irrotational ﬂow represented by the complex potential F(z) on
the (x, y) plane, and that the force acting on B is given by (X, Y ).
By using the expression (5.57) for the pressure p on the body surface
(given by a closed curve C), show the following Blasius’s formula:
X −iY =
1
2
iρ
0

−
γ
2πi
log z, γ = const. (> 0). (5.93)
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5.10. Problems 113
Fig. 5.13. Flow of F
γ
(z).
In addition, compute the velocity circulation around the
cylinder.
(ii) Determine the force (X, Y ) acting on the cylinder C in the ﬂow
F
γ
(z) by using the Blasius formula (5.92).
[The force formula to be obtained is the Kutta–Joukowski’s for-
mula for the lift, Y = ρ
0
Uγ (with X = 0), acting on the cylinder.]
Problem 5.9 Integral formula
Verify the integral formula (5.79) for any constant vectors A and B,
where e is the unit vector in the radial direction, expressed as
e = (e
i
) = (e
x
, e
y
, e
z
) = (sin θ cos φ, sin θ sin φ, cos θ) (5.94)
in the (x, y, z) cartesian frame in terms of the polar angle θ and
azimuthal angle φ of the spherical polar system, and dΩ = sin θ dθdφ.
Problem 5.10 Motion of a sphere in a ﬂuid
According to the formulation in Sec. 5.9, determine the equation of
motion for a sphere of mass m
s
, subject to a force f in an incompress-
ible inviscid ﬂuid, assuming that the ﬂow is irrotational. In addition,
apply it to the motion of a spherical bubble (assuming its density
zero) in water in the gravitational ﬁeld of acceleration g.
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Chapter 6
Water waves and sound waves
Fluid motions are characterized by two diﬀerent elements, i.e. vor-
tices and waves. In this chapter, we consider water waves and sound
waves. There exists a fundamental diﬀerence in character between
the two waves from a physical point of view. The vortices will be
considered in the next chapter.
A liquid at rest in a gravitational ﬁeld is in general bounded above
by a free surface. Once this free surface experiences some disturbance,
it is deformed from its equilibrium state, generating ﬂuid motion.
Then, the deformation propagates over the surface as a wave. Waves
are observed on water almost at any time and are called water waves
which are sometimes called a surface wave. The surface wave is a
kind of dispersive waves whose phase velocity depends on its wave
length.
On the other hand, a sound wave is nondispersive and the phase
velocity of diﬀerent wave lengths take the same value. This results
in a remarkable consequence, i.e. invariance of wave form during
propagation.
6.1. Hydrostatic pressure
Suppose that water of uniform density ρ is at rest in a uniform ﬁeld
of gravity. Then, setting v = 0 and f = g = (0, 0, −g) in (5.2),
with respect to the (x, y, z)-cartesian frame, the z axis taken verti-
cally upward (where g is a constant of gravitational acceleration),
115
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116 Water waves and sound waves
the Euler equation of motion reduces to
grad p = ρg, = ρ (0, 0, −g). (6.1)
Horizontal x and y components and vertical z component of this
equation are
∂p
∂x
= 0,
∂p
∂y
= 0,
∂p
∂z
= −ρg.
Since the density ρ is a constant, we obtain
p = p(z) := −ρgz + const.
Provided that the pressure on the surface is equal to the uniform
value p
0
(the atmospheric pressure) at every point, the surface is
given by z = const., called the horizontal plane. Since the surface
is determined by the pressure solely without any other constraint, it
is also called a free surface.
Let the horizontal free surface be at z = 0, where p = p
0
. Then
we have from the above equation,
p = p
0
−ρgz (z < 0). (6.2)
This pressure distribution is called the hydrostatic pressure
(Fig. 6.1).
Our problem is as follows. Initially, it is assumed that the water
is at rest. Next, a small external disturbance pressure acts on the
surface of water and deforms it. Needless to say, the state of rest is
irrotational since v = 0. Kelvin’s circulation theorem tells us that the
0
Fig. 6.1. Hydrostatic pressure.
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6.2. Surface waves on deep water 117
irrotationality maintains itself in the ﬂuid motion thereafter because
the gravity force is conservative. The deformation propagates as a
wave, and the water motion can be represented by a velocity potential
Φ. It will be found that the wave motion is limited to a neighborhood
of the surface, and the water motion decays rapidly with increasing
depth from the surface. Hence, it is called a surface wave. The sur-
face wave is characterized as a dispersive wave which will be clariﬁed
in the analysis below.
6.2. Surface waves on deep water
It is assumed that the water is inviscid and that its motion is incom-
pressible and irrotational. Then according to Sec. 5.5, the velocity
potential is governed by the Laplace equation. Furthermore for sim-
pliﬁcation, the motion is assumed to be two-dimensional in the x
(horizontal) and z (vertical) plane, uniform in the y (horizontal)
direction. In this case, the velocity is represented as
v = grad Φ(x, z, t) = (u, 0, w) (6.3)
(with t the time), and the velocity potential satisﬁes the following
equation:
∆Φ = Φ
xx
+ Φ
zz
= 0. (6.4)
In addition, it is assumed that the depth is inﬁnite. The surface
deformation is expressed as
z = ζ(x, t). (6.5)
Under the surface, there is an irrotational ﬂuid motion. The Laplace
equation (6.4) is investigated in the domain, −∞ < z < ζ(x, t).
The wave is determined by two boundary conditions imposed on the
surface z = ζ, which are now considered (Fig. 6.2).
6.2.1. Pressure condition at the free surface
This is the condition that the surface z = ζ is acted on by the uniform
atmospheric pressure p
0
. The fact that the ﬂuid motion is irrotational
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118 Water waves and sound waves
Fig. 6.2. Surface wave on deep water.
enables us to use the integral (5.29), which can be written as
Φ
t
+
1
2
q
2
+ (p/ρ) + gz = C,
q
2
= u
2
+ w
2
= Φ
2
x
+ Φ
2
z
.
(6.6)
In Eq. (6.6), p is set equal to p
0
on the surface z = ζ and the constant
C is equal to p
0
/ρ at the initial instant when the water is at rest:
namely p = p
0
at z = 0, Φ
t
= 0, and q
2
= 0. Thus, we obtain the
ﬁrst condition,
BC1 : Φ
t
+
1
2
q
2
+ gζ = 0 at z = ζ(x, t). (6.7)
This is called the dynamic condition.
From (6.6) with C = p
0
/ρ, the pressure within the ﬂuid is given by
p = p
0
−ρ

Φ
t
+
1
2
q
2
+ gz

(z < ζ) (6.8)
This is an extension of the formula (6.2) taking account of additional
terms of ρΦ
t
and
1
2
ρq
2
.
6.2.2. Condition of surface motion
Surface deformation is caused by the motion of ﬂuid particles moving
with velocity (u, w) on the surface. Its mathematical representation
is given as follows. Using a function ζ(x, t) of the surface, we deﬁne
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6.3. Small amplitude waves of deep water 119
a new function f by
f(x, z, t) := z −ζ(x, t). (6.9)
Every point (x, z) on the free surface satisﬁes the equation f(x, z, t) =
0 at any time t. At a time t + ∆t inﬁnitesimally after t, a particle
located at (x, z) at t displaces to (x+u∆t, z +w∆t), and still satis-
ﬁes f(x +u∆t, z +w∆t, t +∆t) = 0. This means that the following
equation holds:
Df
Dt
= 0, or equivalently f
t
+ uf
x
+ wf
z
= 0.
Here, we have f
t
= −ζ
t
, f
x
= −ζ
x
and f
z
= 1 from (6.9). Substituting
these in the above equation, we obtain the second condition,
BC2 : ζ
t
+ uζ
x
= w at z = ζ(x, t). (6.10)
This is called the kinematic condition.
In steady problem, one has the form ζ = ζ(x) and hence
ζ
t
= 0. Then, the above equation (6.10) reduces to dζ/dx = w/u.
This describes simply that the surface coincides with a stream-line, in
other words, ﬂuid particles move over the surface as its stream-line.
Another condition is for the bottom. The ﬂuid motion decays at
an inﬁnite depth:
BC3 : q = |grad Φ| =

u
2
+ w
2
→0 as z →−∞. (6.11)
The three conditions BC1–BC3 are the boundary conditions for the
motion in deep water.
Note that the governing equation (6.4) is linear with respect to Φ.
However, the boundary conditions (6.7) and (6.10) on the free surface
are nonlinear with respect to the velocity and surface deformation.
6.3. Small amplitude waves of deep water
6.3.1. Boundary conditions
It is assumed that the displacement ζ of the free surface and the ﬂuid
velocities u, w are small in the sense described in Sec. 6.3.3 below.
Then one may linearize the problem by neglecting terms of second or
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120 Water waves and sound waves
higher orders with respect to those small quantities. This is called a
linearization. The magnitudes of velocity potential |Φ| is assumed to
be inﬁnitesimally small as well because the water was at rest initially.
Equation (6.4) of Φ is linear originally, hence no term is neglected.
The surface boundary condition (6.7) has a quadratic term q
2
.
Neglecting this, we have
Φ
t
+ gζ = 0 at z = ζ(x, t).
Taylor exansion of the ﬁrst term with respect to z is
Φ
t
(x, ζ, t) = Φ
t
(x, 0, t) + ζ∂
z
Φ
t
(x, 0, t) + O(ζ
2
).
The second term consists of two ﬁrst order terms ζ and ∂
z
Φ
t
, i.e. of
second order. Thus, keeping the ﬁrst term only, we have
Φ
t
+ gζ = 0 at z = 0. (6.12)
Likewise, the boundary condition (6.10) is linearized to
ζ
t
= w (w = Φ
z
) at z = 0. (6.13)
Eliminating ζ from (6.12) and (6.13), we obtain
Φ
tt
+ gΦ
z
= 0 at z = 0. (6.14)
Thus, we have arrived at the following mathematical problem includ-
ing Φ only:
∆Φ = Φ
xx
+ Φ
zz
= 0 (0 > z > −∞), (6.15)
|grad Φ| =

(Φ
x
)
2
+ (Φ
z
)
2
→0 as z →−∞, (6.16)
together with the surface condition (6.14).
This is a system of equations (6.14)–(6.16) which determine water
waves of small amplitude. Once Φ(x, z, t) is found, the wave form
ζ(x, t) is determined by (6.12).
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6.3. Small amplitude waves of deep water 121
6.3.2. Traveling waves
Let us try to ﬁnd a solution of the following form,
Φ = f(z) sin(kx −ωt), (6.17)
(k, ω: constants, assumed positive for simplicity), which represents
a sinusoidal wave of the free surface traveling in the x direction
with phase velocity ω/k. The constant k is called the wavenumber
and ω the angular frequency, and λ = 2π/k is the wavelength
(Fig. 6.3). The amplitude f(z) is a function of z to be determined.
Substituting (6.17) in the Laplace equation (6.15), and dropping
oﬀ the common factor sin(kx −ωt), we obtain
f

=
1
2
k
2
B
2
=
1
2
ω
2
A
2
,
which was neglected. On the other hand, the two terms in the bound-
ary condition (6.7) are of the form ∓gAcos(kx − ωt). Hence, the
condition by which the q
2
term can be neglected is
1
2
ω
2
A
2
gA
=
1
2
kA = π
A
λ
1
1
This must be discriminated from the gravity wave in the general relativity theory
for gravitation.
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6.3. Small amplitude waves of deep water 123
from (6.20) where λ = 2π/k. Hence, the condition for the linear
approximation to be valid is that the wave amplitude A is much less
than the wavelength λ.
In the kinematic condition (6.10), a term of form uζ
x
was
neglected. Its magnitude is of order k
2
AB = kωA
2
from (6.21) and
(6.23). On the other hand, the linear terms ζ
t
and w retained are
of form ωAsin(kx − ωt). The condition for linear approximation to
be valid is still the same as before, kA 1, stating that the wave
amplitude should be much smaller than the wavelength.
6.3.4. Particle trajectory
Let us consider the trajectories of ﬂuid particles moving in the trav-
eling wave (6.21). Suppose that a particle position is denoted by
(X(t), Z(t)). Assuming that its deviation from the mean position
(x
0
, z
0
) is small, the variables x and z in the expressions of u and w
are replaced by x
0
and z
0
, respectively. Then the equation of motion
of the particle is given by
dX
dt
= u(x
0
, z
0
, t),
dZ
dt
= w(x
0
, z
0
, t).
Using (6.23) and kB = ωA, these equations can be integrated, giving
X(t) −x
0
= −Ae
kz
0
sin(kx
0
−ωt),
Z(t) −z
0
= Ae
kz
0
cos(kx
0
−ωt).
(6.24)
It is found that the particle moves around a circle centered at (x
0
, z
0
)
and that its radius Ae
kz
0
decays rapidly as the depth |z
0
|(z
0
< 0)
increases. Most part of the kinetic energy is included within the
depth of a fourth of a wavelength under the surface (|z
0
|/
1
4
λ = k|z
0
|/
1
2
π < 1).
6.3.5. Phase velocity and group velocity
The surface elevation of the form ζ = Acos(kx − ωt) represents a
traveling wave. This is understood from the phase kx − ωt of the
cos-function. Requiring that the phase is equal to a ﬁxed value φ
0
,
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124 Water waves and sound waves
we have kx −ωt = φ
0
, from which we obtain

dx
dt

φ
0
=const.
=
ω
k
=: c
p
. (6.25)
This means that the same value of elevation ζ
0
= Acos φ
0
moves
with the speed c
p
= ω/k in the x direction. The velocity c
p
is termed
the phase velocity. In the case of water waves, we have another
important velocity. That is, group velocity c
g
is deﬁned by the
following:
c
g
:=
dω
dk
. (6.26)
In general, wave motion is characterized by a general dispersion
relation,
ω = ω(k). (6.27)
This functional relation immediately gives c
p
and c
g
deﬁned above.
In water waves characterized by the dispersion relation (6.20), we
have
ω(k) =

g/k =
1
2
c
p
. (6.30)
The two velocities are interpreted as follows. The phase velocity
denotes the moving speed of a phase φ
0
of a component with
wavenumber k and frequency ω. On the other hand, the group
velocity c
g
(k) is given various interpretations of diﬀerent physical
signiﬁcance:
(a) Speed of a wave packet, a group of waves having a wavenumber k
0
and wavenumbers of its immediate neighborhood (Problem 6.3).
(b) Speed of transport of wave energy of wavenumber k.
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6.4. Surface waves on water of a ﬁnite depth 125
(c) Suppose that initial disturbance is localized spatially. Then,
dω/dk is the traveling speed of k component characterized by
wavelength λ
k
= 2π/k.
It is remarkable that waves of larger wavelength λ moves faster than
shorter ones in deep water, according to (6.29) of phase velocity.
Suppose that the initial wave is composed of wave components of
various wavelengths. Because the phase velocities are diﬀerent for
diﬀerent wavelengths, each component of the wave moves with diﬀer-
ent speeds. Therefore, the total wave form obtained by superposition
of those components deforms in the course of time. This indicates
that the waves disperse. Namely, the waves are dispersive if the
phase velocity depends on the wavenumber k.
Water wave is a familiar example of dispersive waves. An example
of nondispersive waves is the sound wave, which is investigated in
Sec. 6.6.
6.4. Surface waves on water of a ﬁnite depth
Here we consider surface gravity waves at a ﬁnite depth of water, a
more realistic problem. The bottom is assumed to be horizontal at
a depth h (Fig. 6.4). The boundary condition at the bottom is given
by vanishing normal velocity,
BC3

,
(6.36)
since we have tanh kh = kh −
1
3
(kh)
3
+· · · for small kh.
Namely, the wave speed is given approximately as
√
gh in shallow
water, and becomes slower as the depth decreases. This property is
applied to explain the fact that crests of sea water wave near a coast
become parallel to the coast line as the waves approach a coast, because
the wave speed slows down according to
√
gh, as the depth decreases.
2
6.5. KdV equation for long waves on shallow water
John Scott Russel observed large solitary waves along canals between
Glasgow and Edinburgh (or and Ardrossan) of Scotland in 1834, which
are now recognized as the ﬁrst observation of solitary waves, called
the soliton. One day (the happiest to him), something unexpected
2
Lagrange (1782) solved Eq. (6.15) for water waves at shallow water of depth h, and
obtained the dispersion relation ω
2
= gk tanh kh and the phase speed c
p
=
√
gh on
the basis of the boundary conditions (6.12) and (6.14). Laplace also obtained
ω
2
= gk tanhkh earlier in 1776 by solving the Laplace equation (6.15) under a
diﬀerent formulation [Dar05].
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6.5. KdV equation for long waves on shallow water 127
happened. He was on a vessel moving at a high velocity in order to
understand an anomalous decrease of resistance as a young engineer of
naval architecture. He observedthat alarge wave was generatedwhenit
stopped suddenly. He immediately left the vessel and got on horseback.
Thewavepropagatedalongdistancewithout changeof its form. Hethen
conﬁrmed it was in fact a large, solitary, progressive wave [Dar05].
Later, both Boussinesq (1877) and Korteweg and de Vries (1895)
(apparently unaware of the Boussinesq’s study [Dar05]) succeeded
in deriving an equation allowing stationary advancing waves without
change of form, i.e. solutions which do not show breakdown at a ﬁnite
time. In the problem of long waves in a shallow water channel of depth
h
∗
, it is important to recognize that there are two dimensionless
parameters which are small:
α =
a
∗
h
∗
, β =

h
∗
λ

2
, (6.37)
where a
∗
is a wave amplitude and λ is a representative horizontal
scale characterizing the wave width.
In order to derive the equation allowing permanent waves (trav-
eling without change of form), it is assumed that α ≈ β 1. Per-
forming a systematic estimation of order-of-magnitudes under such
conditions, one can derive the following equation,
∂
τ
u +
3
2
u∂
ξ
u +
1
6
∂
3
ξ
u = 0 (6.38)
(see [Ka04, Ch. 5 & App. G] for its derivation), where
ξ =

α
β

1/2
x −c
∗
t
λ
, τ =

α
3
β

1/2
c
∗
t
λ
. (6.39)
The function u(x, t) denotes not only the surface elevation normalized
by a
∗
, but also the velocity (normalized by ga
∗
/c
∗
),
u = dx
p
/dt, (6.40)
of the water particle with its location at x = x
p
(t).
One of the characteristic features is the existence of the third
order derivative term
1
6
∂
3
ξ
u in Eq. (6.38). A signiﬁcance of this term
is interpreted as follows. Linearizing Eq. (6.38) with respect to u,
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128 Water waves and sound waves
we obtain ∂
τ
u + α∂
3
ξ
u = 0

where α =
1
6

. Assuming a wave form
u
w
∝ exp[i(ωτ − kξ)] (the wavenumber k and frequency ω) and
substituting it, we obtain a dispersion relation, ω = −αk
3
. Phase
velocity is deﬁned as c(k) := ω/k = −αk
2
. Namely, a small amplitude
wave u
w
propagates with the nonzero speed c(k) = −αk
2
, and the
speed is diﬀerent at diﬀerent wavelengths (= 2π/k). This eﬀect was
termed as wave dispersion in Sec. 6.3. What is important is that the
new term takes into account the above wave propagation, in addition
to the particle motion dx
p
/dt (a physically diﬀerent concept).
Replacing u by v =
3
2
u, we obtain
∂
τ
v + v∂
ξ
v +
1
6
∂
3
ξ
v = 0. (6.41)
This equation is now called the KdV equation after Korteweg and
de Vries (1895). Equation (6.41) allows steady wave solutions, which
they called the permanent wave. Setting v = f(ξ −bτ) (b: a constant)
and substituting it into (6.41), we obtain f

+ 6ﬀ

−6bf

= 0. This
can be integrated twice. Choosing two integration constants appro-
priately, one ﬁnds two wave solutions as follows:
v = A sech
2
¸

ξ
f(ξ)dξ is an arbitrary function of the variable ξ
only, while f
2
(η) is an arbitrary function of η only. Both are to be
determined by initial condition or boundary condition. It is readily
veriﬁed that this satisﬁes Eq. (6.61). Same expressions can be given to
u and p
1
, too. This can be shown by noting that these are connected
to ρ
1
with (6.55) and (6.57) (or (6.58)).
The solution just found can be easily understood as follows.
Regarding the ﬁrst term f
1
(x − ct), if the time t is ﬁxed, f
1
is a
function of x only, which represents a spatial wave form. Next, if the
coordinate x is ﬁxed, f
1
is a function of t only, which represents a
time variation of ρ
1
. Furthermore, it is important to recognize that
the value of function f
1
is unchanged if the value x−ct has the same
value for diﬀerent values of x and t.
For example, suppose that x is changed by ∆x = c∆t when the
time t advanced by ∆t, then we have x + ∆x − c(t + ∆t) = x − ct.
Hence the value of f
1
is unchanged. This means that the value of
f
1
is moving forward with the velocity ∆x/∆t = c in the positive
direction of x. Namely, the wave form f
1
(x −ct) is moving with the
velocity c at all points on x since c is a constant.
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134 Water waves and sound waves
This is understood to mean that the function f
1
(x−ct) represents
propagation of a wave of speed c, which is the sound speed.
Likewise, it is almost evident that the second term f
2
(x + ct)
represents propagation of a wave in the negative x direction (∆x =
−c∆t). Thus, it is found that Eq. (6.59) or (6.60) is a diﬀerential
equation representing waves propagating in the positive or negative x
directions with the speed c. The wave is called longitudinal because
the variable u expresses ﬂuid velocity in the same x direction as the
wave propagation.
Provided that the propagation direction is chosen as that of posi-
tive x, the velocity, density and pressure are related by simple useful
relations. Suppose that
ρ
1
= f(x −ct).
In view of ∂ρ
1
/∂x = f

The imaginary part also satisﬁes the wave equation.
Such a type of solution as (6.66) characterized by a single fre-
quency ω is called a monochromatic wave, according to optics. Fur-
thermore, if the solution (6.67) is substituted, it is found that u of
(6.66) becomes
u(x, t) = A exp[i(ω/c)(x −ct)] + B exp[−i(ω/c)(x + ct)],
which has the form of (6.62).
4
See (5.49) for the deﬁnition of a complex number z by z = re
iθ
.
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136 Water waves and sound waves
Setting A = ae
iα
(where a, α are real constants) and taking the
real part of the ﬁrst term of the above expression (assuming B = 0),
we obtain
u
r
(x, t) = a cos(kx −ωt + α) = a cos φ, (6.68)
k = ω/c, φ := kx −ωt + α (6.69)
where a is the wave amplitude, and φ the phase of the wave. In
addition, k = ω/c is the wavenumber. This is due to the property
that the phase φ of the above wave increases by 2πk when x increases
by 2π. That is, the number of wave crests is k in the x-interval 2π.
The wavelength is given by λ = 2π/k.
The wave we have investigated so far is regarded as a plane wave
in the (x, y, z) three-dimensional space, because x = const. describes
a plane perpendicular to the x axis. The wave of the form (6.68) is
called a monochromatic plane wave.
A complex monochromatic plane wave is given by
u
k
(x, t) = A
k
exp[i(kx −ωt)] = A
k
exp[ik(x −ct)]. (6.70)
The theory of Fourier series and Fourier integrals places a particular
signiﬁcance at this representation, because arbitrary waves are rep-
resented by superposition of monochromatic plane waves of diﬀerent
wavenumbers. So, the monochromatic plane waves are regarded as
Fourier components, or spectral components.
The relation (6.69) between the wavenumber k and frequency ω
is the dispersion relation considered in Sec. 6.3.5. Using the above
dispersion relation k = ω/c of the sound wave, we can calculate the
phase velocity c
p
and the group velocity c
g
:
c
p
=
ω
k
= c, c
g
=
dω
dk
= c.
Namely, in the sound wave, the phase velocity is the same at all
wavelengths, and in addition, it is equal to the group velocity. Owing
to this property, the wave form of sound waves is unchanged during
propagation. This property is an advantage if it is used as a medium
of communication. Interestingly, light has the same nondispersive
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6.7. Shock waves 137
property, and light speed is the same for all wavelengths in vacuum
space.
6.7. Shock waves
Shock waves are formed around a body placed in a supersonic ﬂow,
or formed by strong impulsive pressure increases which act on a sur-
face surrounding a ﬂuid. Shocks are sometimes called discontinuous
surfaces because velocity, pressure and density change discontinu-
ously across the surface of a shock wave. So far, we have considered
continuous ﬁelds which are represented by continuous diﬀerentiable
functions of position x and time t, and the governing equations are
described by partial diﬀerential equations. Let us investigate what is
the circumstance when discontinuity is allowed.
Fluid motions are governed by three conservation laws of mass,
momentum and energy considered in Chapter 3. These are basic
constraints to be satisﬁed by ﬂuid motions. Even when there exists a
discontinuous surface, if these conservation laws are not violated, it
should be allowed to exist physically. For example, mass conservation
is satisﬁed if the rate of inﬂow of ﬂuid into the discontinuous surface
from one side is the same as the rate of outﬂow from the other side
(Fig. 6.7).
Fig. 6.7. Discontinuous surface.
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138 Water waves and sound waves
In the one-dimensional problem considered in Sec. 6.6.1, the dis-
continuous surface should be a plane perpendicular to the x axis.
Suppose that a discontinuous surface S, i.e. a shock wave, is at rest
at position x = x
0
(Fig. 6.7), and that the ﬂuid is ﬂowing into the
surface S from the left with velocity u
1
and ﬂowing out of S to the
right with velocity u
2
. It is assumed that the states on both sides of
S are steady and uniform, and that the density and pressure are ρ
1
and p
1
on the left and ρ
2
and p
2
on the right, respectively. The mass
ﬂux into the surface S from the left is ρ
1
u
1
per unit time and per unit
area of S. The outﬂux is expressed by ρ
2
u
2
. Therefore, the mass con-
servation is given by ρ
1
u
1
= ρ
2
u
2
. Precisely speaking, the equation
must be considered on the basis of the conservation equations.
Because the problem under consideration is steady, the time
derivative term ∂
t
in the equation of mass conservation (6.44) van-
ishes. Then we have ∂
x
(ρu) = 0, giving ρu = const. This leads to
ρ
1
u
1
= ρ
2
u
2
. (6.71)
This is equivalent to the relation obtained in the above consideration.
Similarly, neglecting the time derivative terms in the conserva-
tion equations of momentum (6.45) and energy (6.46), we obtain the
following:
ρ
1
u
2
1
+ p
1
= ρ
2
u
2
2
+ p
2
, (6.72)
1
2
u
2
1
+
c
2
1
γ −1
=
1
2
u
2
2
+
c
2
2
γ −1
(6.73)
where the enthalpy h has been replaced by the expression of an ideal
gas, h = c
2
/(γ −1), given as (C.6)
Introducing J by
J = ρ
1
u
1
= ρ
2
u
2
, (6.74)
Eq. (6.72) leads to the following two expressions:
J
2
= (p
2
−p
1
)
ρ
1
ρ
2
ρ
2
−ρ
1
, (6.75)
u
1
−u
2
=
p
2
−p
1
J
=

. (6.77)
These relations connecting the two states of upstream and down-
stream of a discontinuity surface are called Rankin–Hugoniot’s
relation (Rankin (1870) and Hugoniot (1885)), known as the shock
adiabatic.
For given ρ
1
and p
1
, the relations (6.76) and (6.77) determine ρ
2
and p
2
with the help of the relation c
2
= γp/ρ of (C.2), depending
on the parameters J and γ.
6.8. Problems
Problem 6.1 One-dimensional ﬁnite amplitude waves
We consider one-dimensional ﬁnite amplitude waves on the basis of
the continuity equation (6.47) and the equation of motion (6.48).
(i) Using the isentropic relation (6.55) and rewriting (6.47) in terms
of pressure p and velocity u, derive the following system of
equations:
∂
t
u +
1
ρc
∂
t
p + (u + c)

∞
−∞
A(k)e
i(kx−ωt)
dk. (6.83)
Consider a wavemaker which oscillates at a single frequency ω
0
. Its
amplitude ﬁrst increased from zero to a maximum and then returned
to zero again, slowly with a time scale much larger than the oscil-
lation period 2π/ω
0
. By this wave excitation, it is found that most
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6.8. Problems 141
of the wave energy is concentrated on a narrow band of wavenum-
bers around k
0
. Hence, the dispersion relation is approximated by
the following linear relation:
ω(k) = ω
0
+ c
g
(k −k
0
), ω
0
= ω(k
0
), c
g
= dω/dk. (6.84)
(The amplitude A(k) is regarded as zero for such k-values in which
the above linear relation loses its validity.)
(i) Show that the resulting wave would be given by the following
form of a wave packet, with ξ = x −c
g
t:
ζ(x, t) = F(ξ)e
i(k
0
x−ω
0
t)
. (6.85)
In addition, write down Fourier representation of the amplitude
function F(ξ).
(ii) When the Fourier amplitude is A(k) = A
0
exp[−a(k − k
0
)
2
], a
Gaussian function around k
0
, give an explicit form of the wave
packet.
The function F(ξ) is an envelope moving with the group velocity c
g
and enclosing carrier waves e
i(k
0
x−ω
0
t)
within it (Fig. 6.9).
Fig. 6.9. Wave packet and group velocity c
g
.
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Chapter 7
Vortex motions
Vortex motions are vital elements of ﬂuid ﬂows. In fact, most dynam-
ical aspects of ﬂuid motions are featured by vortex motions. Already
we considered some motions of vortices: vortex sheet and shear layer
[Sec. 4.6.2(b)], vortex line and point vortex (Secs. 5.6.5 and 5.8.5), etc.
Analytical study of vortex motions is mostly based on vortic-
ity which is deﬁned as twice the angular velocity of local rotation
(Sec. 1.4.3). The vorticity equation is derived in Sec. 4.1 from the
equation of motion. It is to be remarked that the no-slip condition
and the boundary layer are identiﬁed as where vorticity is generated
(Sec. 4.5). In Chapter 12, it will be shown that vorticity is in fact a
gauge ﬁeld associated with rotational symmetry of the ﬂow ﬁeld, on
the basis of the gauge theory of modern theoretical physics.
7.1. Equations for vorticity
7.1.1. Vorticity equation
Equation of the vorticity ω = curl v was given for compressible ﬂows
in Sec. 3.4 in an inviscid ﬂuid (ν = 0) as
∂
t
ω + curl (ω ×v) = 0. (7.1)
For viscous incompressible ﬂows, two equivalent equations given in
Sec. 4.1 are reproduced here:
∂
t
ω + curl (ω ×v) = ν∇
2
ω, (7.2)
∂
t
ω + (v · ∇)ω = (ω · ∇)v +ν∇
2
ω. (7.3)
143
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144 Vortex motions
7.1.2. Biot–Savart’s law for velocity
Let us consider a solenoidal velocity ﬁeld v(x) (deﬁned by div v = 0,
i.e. incompressible) induced by a compact vorticity ﬁeld ω(x). That
is, the vorticity vanishes out of a bounded open domain D:
ω(x)

= 0, for x ∈ D
= 0, for x out of D.
(7.4a)
It is to be noted in this case that the following space integral of each
component ω
k
(y) vanishes for k = 1, 2, 3:

D
ω(y, t)
|x −y|
d
3
y, (7.5a)
where y denotes a position vector for the integration (with t ﬁxed).
It can be shown that curl v = ω in the free space for the compact
vorticity ﬁeld ω(x) of (7.4a) (Problem 7.1).
The velocity v is derived from the vector potential A by taking
its curl:
v(x) = curl A = −
1
4π

(x −y) ×ω(y)
|x −y|
3
d
3
y, (7.5b)
where the formula ∇×(a/|x|) = a ×x/|x|
3
(a is a constant vector)
is used. This gives the velocity ﬁeld when the vorticity distribution
ω(y) is known, called the Biot–Savart law. Originally, this law was
given for a magnetic ﬁeld (in place of v) produced by steady electric
current (in place of ω) of volume distribution in the electromagnetic
theory.
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7.1. Equations for vorticity 145
As |x| →∞, the velocity (7.5b) has an asymptotic behavior:
v(x) = −
1
4π|x|
3
x ×

ω(y)d
3
y +O(|x|
−1
)

= O(|x|
−3
), (7.5c)
owing to (7.4b).
7.1.3. Invariants of motion
When the ﬂuid is inviscid and of constant density and the vorticity
is governed by the vorticity equation (7.1), we have four invariants
of motion:
Energy : K =
1
2

V
∞
v
2
d
3
x (7.6)
=
1
2

D
ω(x, t) · A(x, t) d
3
x (7.7)
=

D
v · (x ×ω) d
3
x, [Lamb32, Sec. 153]
(7.8)
Impulse : P =
1
2

D
x ×ω(x, t) d
3
x, (7.9)
Angular impulse : L =
1
3

D
x ×(x ×ω) d
3
x, (7.10)
Helicity : H =

D
v · ωd
3
x, (7.11)
where div v = 0, and V
∞
is unbounded space of inviscid ﬂuid ﬂow.
Here, we verify invariance of the total kinetic energy K for ﬂuid
motion under the compact vorticity distribution (7.4a). Alternative
expressions of P and L are given in the end.
To show K = const. for an inviscid ﬂuid under ∂
k
v
k
= 0 and
(7.4a), we take scalar product of v
i
and (3.14) with ρ = const. and
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146 Vortex motions
f
i
= 0, and obtain
∂
t

V
∞
(∇×v) · Ad
3
x,
by omitting integrated terms since A = O(|x|
−2
) obtained by
the same reasoning as in the case of (7.5c). This is (7.7) since
∇×v = ω.
In order to show (7.8), ﬁrst note the following identity:
(x ×ω)
i
= (x ×(∇×v))
i
= x
k
∂
i
v
k
−x
k
∂
k
v
i
= 2v
i
+∂
i
(x
k
v
k
) −∂
k
(x
k
v
i
), (7.5e)
where ∂
i
x
j
= δ
ij
and ∂
j
x
j
= 3 are used to obtain the last equality,
and in the second equality of the ﬁrst line the vector identity (A.18)
is applied. Hence, the integrand of (7.8) is given by
v · (x ×ω) = v
i
(x ×ω)
i
= 2v
2
+v
i
∂
i
(x
k
v
k
) −v
i
∂
k
(x
k
v
i
)
=
1
2
v
2
+∂
i
(x
k
v
k
v
i
) −∂
k

D
|x|
2
ωd
3
x. (7.4g)
Problem 7.2 questions how to verify invariance of P, L and H of
(7.9)–(7.11).
7.2. Helmholtz’s theorem
7.2.1. Material line element and vortex-line
In an inviscid incompressible ﬂuid where ν = 0 and div v = 0, the
vorticity equation (7.3) reduces to
D
Dt
ω = (ω · ∇)v, (7.12)
where Dω/Dt = ∂
t
ω + (v · ∇)ω. In order to see the meaning of this
equation, we consider the motion of an inﬁnitesimal line element.
We choose two ﬂuid particles located at suﬃciently close points
x and x + δs
a
at an instant, δs
a
denoting the material line element
connecting the two points. Rate of change of the line element vector
δs
a
is given by the diﬀerence of their velocities, that is
D
Dt
δs
a
= v(x +δs
a
) −v(x) = (δs
a
· ∇)v + (|δs
a
|
2
).
Neglecting the second and higher order terms, we have
D
Dt
δs
a
= (δs
a
· ∇)v. (7.13)
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148 Vortex motions
The similarity between this and (7.12) is obvious. Taking the line
element δs
a
= |δs
a
|e parallel to ω(x) = |ω|e at a point x where e is
a unit vector in the direction of ω, the right-hand sides of (7.12) and
(7.13) can be written as |ω|(e · ∇)v and |δs
a
|(e · ∇)v, respectively.
Hence, we obtain the following equation:
1
|ω|
D
Dt
ω =
1
|δs
a
|
D
Dt
δs
a
, (7.14)
since both terms are equal to (e · ∇)v. Thus, it is found that relative
rate of change of ω following the ﬂuid particle is equal to relative
rate of stretching of a material line element parallel to ω.
7.2.2. Helmholtz’s vortex theorem
Regarding the total ﬂux of vortex-lines passing through a material
closed curve, Kelvin’s circulation theorem of Sec. 5.3 states that it
is invariant with time for the motion of an ideal ﬂuid of homentropy
(subject to a conservative external force). This is valid for ﬂows of
compressible ﬂuids.
For the case of variable density ρ, the vorticity equation is already
given by (3.33):
D
Dt

ω
ρ

=

ω
ρ
· ∇

v, (7.15)
which reduces to (7.12) when ρ = const. This equation states that
ω/ρ behaves like a material line element parallel to ω.
For the vortex motion governed by Eq. (7.15), one can deduce
the following laws of vortex motion, originally given by Helmholtz
(1858):
(i) Fluid particles initially free of vorticity remain free of vorticity thereafter.
(ii) The vortex-lines move with the ﬂuid. In other words, ﬂuid particles on a
vortex-line at any instant will be on the vortex-line at all subsequent times.
(iii) Strength of an inﬁnitesimal vortex tube deﬁned by |ω|θ does not vary with
time during the motion, where θ is the inﬁnitesimal cross-section of the
vortex tube.
This is called the Helmholtz’s vortex theorem derived as follows.
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7.2. Helmholtz’s theorem 149
Multiplying a constant λ on both sides of (7.13) and subtracting
it from (7.15), we obtain
D
Dt

ω
ρ
−λδs
a

=

ω
ρ
−λδs
a

· ∇v. (7.16)
We choose a vortex-line, which is a space curve x
σ
= (x(σ),
y(σ), z(σ)) parameterized with a variable σ. Suppose that the
line element δs
a
coincides with a local tangent line element at s,
deﬁned by
δs
a
≡ δs
σ
= (x

σ
,
where v = (u, v, w). Let us investigate the evolution of the system
by following the material particles on the vortex-line with a speciﬁed
value of σ and dσ.
Equation (7.17) is a system of ordinary diﬀerential equations for
ξ(t), η(t) and ζ(t). Suppose that we are given the initial condition,
vortex-line
Fig. 7.1. A vortex-line (x(σ), y(σ), z(σ)) and a material line element δs
a
.
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150 Vortex motions
such that
X
σ
(0) = (ξ(0), η(0), ζ(0)) = (0, 0, 0). (7.18)
The above equations (7.17) give the (unique) solution,
X
σ
(t) = 0, at all t > 0, (7.19)
for usual smooth velocity ﬁelds v(x).
1
Hence, we obtain the relation,
ω(x
σ
, t) = λρ(x
σ
, t)δs
σ
(t), (7.20)
for all subsequent times t. Therefore, the vortex-line moves with the
ﬂuid line element, and the material line element δs
σ
forms always
part of a vortex-line at x
σ
. This is the Helmholtz’s law (ii).
2
Thus, it is found that the length δs = |δs
σ
| of the line element
will vary as ω/ρ (since ω = λρδs), where ω = |ω|. Using the cross-
section θ of an inﬁnitesimal vortex tube, the product ρδsθ denotes
the ﬂuid mass of the line element which must be conserved during the
motion. Since the product ρδs is proportional to ω, we obtain that
the strength of the vortex deﬁned by the product ωθ is conserved
during the motion. This veriﬁes the Helmholtz’s law (iii).
Regarding the law (i), we set λ = 0 since λ was an arbitrary
constant parameter. Then, the above problem (7.18) and (7.19) states
that a ﬂuid particle initially free of vorticity remains free thereafter,
verifying the Helmholtz’s ﬁrst law.
7.3. Two-dimensional vortex motions
Two-dimensional problems are simpler, but give us some useful infor-
mation. Let us consider vortex motions of an inviscid ﬂuid governed
1
Mathematically, uniqueness of the solution is assured by the Lipschitz condition
for the functions on the right-hand sides of (7.17), which should be satisﬁed
by usual velocity ﬁelds u(x), v(x) and w(x). See the footnote in Sec. 1.3.1. In
[Lamb32, Sec. 146], the solution X
σ
(t) = 0 was given by using the Cauchy’s
solution.
2
In [Lamb32, Sec. 146], it is remarked that the Helmholtz’s reasoning was not
quite rigorous. In fact, his equation resulted in the form (7.16), but there was no
term on the right-hand side.
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7.3. Two-dimensional vortex motions 151
Fig. 7.2. Vorticity in a two-dimensional domain D: (a) Continuous distribution,
(b) discrete distribution.
by (7.12). Suppose that the vorticity is given in an open domain D
of (x, y) plane and vanishes out of D [Fig. 7.2(a)]:
ω = ω(x, y), (x, y) ∈ D
ω = 0, (x, y) ∈ ∂D, out of D.
(7.21)
It is assumed that the vorticity ω tends to zero continuously toward
the boundary ∂D of D. Discrete vorticity distribution like the delta-
function of a point vortex is regarded as a limit of such continuous
distributions [Fig. 7.2(b)].
7.3.1. Vorticity equation
Assuming that the ﬂuid motion is incompressible, the x, y compo-
nents of the ﬂuid velocity are expressed by using the stream function
Ψ(x, y, t) as
u = ∂
y
Ψ, v = −∂
x
Ψ, (7.22)
(Appendix B.2). The z component of the vorticity is given by
ω = ∂
x
v −∂
y
u = −∂
2
x
Ψ−∂
2
y
Ψ = −∇
2
Ψ, (7.23)
where ∇
2
= ∂
2
x
+∂
2
y
. The x, y components of vorticity vanish iden-
tically because w = 0 and ∂
z
= 0, where w is the z component of
velocity and the velocity ﬁeld is independent of z.
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152 Vortex motions
The vorticity equation (7.12) becomes
D
Dt
ω = 0, (7.24)
since (ω · ∇)v = ω∂
z
v = 0. This is rewritten as
∂
t
ω +u∂
x
ω +v∂
y
ω = 0, (7.25)
which is the vorticity equation in the two-dimensional problem.
If the function ω(x, y) is given, Eq. (7.23) is the Poisson equation
∇
2
Ψ = −ω(x, y) of the function Ψ(x, y). Its solution is expressed by
the following integral form,
Ψ(x, y) = −
1
4π

ω dxdy. (7.31)
It can be shown that these four integrals are constant during the
vortex motion. The ﬁrst Γ is the total amount of vorticity in D, which
is also equal to the circulation along any closed curve enclosing the
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7.3. Two-dimensional vortex motions 153
vortex domain D. The Kelvin’s circulation theorem (Sec. 5.2) assures
that Γ is invariant. Below, this is shown directly by using (7.27).
From the three integrals, Γ, X and Y , we deﬁne two quantities by
X
c
=

D
xω dxdy

D
ω dxdy
, Y
c
=

D
yω dxdy

D
ω dxdy
. (7.32)
These are other invariants if Γ, X and Y are invariants. An interesting
interpretation can be given for X
c
and Y
c
. If we regard ω as a hypo-
thetical mass density, then Γ corresponds to the total mass included
in D, and the pair (X
c
, Y
c
) corresponds to the center of mass, in
other words, the center of vorticity distribution with a weight ω.
In order to verify that Γ, X and Y are invariant (see Problem 7.3
for the invariance of R
2
), let us ﬁrst derive the expressions of the
velocities u and v by using the stream function (7.26):
u(x, y) = ∂
y
Ψ =
1
2π

dxdy, (7.35)
where we used the continuity ∂
x
u +∂
y
v = 0 and (7.27).
Integrating the ﬁrst and second terms of (7.35) (a divergence of
the vector x
α
ωv), we obtain the contour integral of x
α
ωv
n
along the
boundary ∂D by the Gauss theorem where ω = 0 (v
n
is the normal
component). Hence, they vanish. The last term vanishes as well. It
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154 Vortex motions
is obvious when α = 0. For α = 1, substituting (7.33), we obtain
dX
1
dt
=

D
ωudA =
1
2π

D

D
(y

−y)
ω(x, y)ω(x

, y

)
(x

−x)
2
+ (y

−y)
2
dAdA

where dA = dxdy and dA

= dx

dy

. The last integral is a four-
fold integral with respect to two pairs of variables, dxdy and dx

dy

.
Interchanging the pairs, the ﬁrst factor (y

−y) changes its sign (anti-
symmetric), whereas the second factor of the integrand is symmetric.
The integral itself should be invariant with respect to the interchange
of (x, y) and (x

, y

). Therefore, the integral must be zero. Hence,
dX/dt = 0. Similarly, one can show dY/dt = 0. Thus, it is veriﬁed
that the three integrals Γ, X and Y are invariant.
7.3.3. Velocity ﬁeld at distant points
Let us consider the velocity ﬁeld far from the vortex domain D. We
choose a point (x

≈
Γ
2π
x
x
2
+y
2
, (7.37)
where Γ is the total vorticity included in the domain D deﬁned by
(7.28), and called the vortex strength of the domain.
7.3.4. Point vortex
When the vortex domain D of Fig. 7.2(a) shrinks to a point P =
(x
1
, y
1
) by keeping the vortex strength Γ to a ﬁnite value k, then we
have the following relation,

limD→P
ω(x, y) dxdy = k.
This implies that there is a concentrated vortex at P, and that the
vorticity can be expressed by the delta function (see Appendix A.7) as
ω(x, y) = kδ(x −x
1
) δ(y −y
1
), (7.38)
Substituting this to (7.33) and (7.34), the velocities are
u(x, y) = −
k
2π
y −y
1
(x −x
1
)
2
+ (y −y
1
)
2
, (7.39)
v(x, y) =
k
2π
x −x
1
(x −x
1
)
2
+ (y −y
1
)
2
, (7.40)
This is the same as the right-hand sides of (7.36) and (7.37) with Γ
replaced by k, and also the expressions (5.47) and (5.48) (and also
(5.73)) if x and y are replaced by x −x
1
and y −y
1
, respectively.
From the deﬁnitions of the integral invariants (7.28)–(7.30) and
(7.32), we obtain Γ = k, X
c
= x
1
and Y
c
= y
1
. Thus, we have the
following.
The strength k of the point vortex is invariant. In addition,
the position (x
1
, y
1
) of the vortex does not change.
Namely, the point vortex have no self-induced motion. In other words,
a rectilinear vortex does not drive itself.
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156 Vortex motions
To represent two-dimensional ﬂows of an inviscid ﬂuid including
point vortices, the theory of complex functions is known to be a
powerful tool. A vortex at the origin z = 0 is represented by the
following complex potential (Sec. 5.8.5),
F
v
(z) =
k
2πi
log z (k : real). (7.41)
The velocity ﬁeld is given by dF
v
/dz = w = u −iv, and we obtain
u = −
k
2πr
sin θ = −
k
2π
y
r
2
, v =
k
2πr
cos θ =
k
2π
x
r
2
, (7.42)
where r
2
= x
2
+ y
2
. It is seen that these are equivalent to (7.39)
and (7.40).
7.3.5. Vortex sheet
Vortex sheet is a surface of discontinuity of tangential velocity
[Fig. 4.5(a)]. Suppose that there is a surface of discontinuity at y = 0
of a ﬂuid ﬂow in the cartesian (x, y, z) plane, and that the velocity
ﬁeld is as follows:
v =

1
2
U, 0, 0

for y < 0 ; v =

−
1
2
U, 0, 0

for y > 0. (7.43)
(Directions of ﬂows are reversed from Fig. 4.5(a).) The vorticity of
the ﬂow is represented by
ω = (0, 0, ω(y)), ω = U δ(y), (7.44)
where δ(y) is the Dirac’s delta function (see A.7). This can be con-
ﬁrmed by using the formula (5.20) (Problem 7.4).
7.4. Motion of two point vortices
First, we consider a system of two point vortices by using the complex
potential (7.41) (see Sec. 5.8.5) on the basis of the theory of complex
functions (Appendix B), and derive the invariants directly from the
equation of motion. After that, we learn that a system of N point
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7.4. Motion of two point vortices 157
vortices (of general number N) is governed by Hamiltonian’s equation
of motion.
Suppose that we have two point vortices which are moving under
mutual interaction, and that their strengths are k
1
and k
2
, and
positions in (x, y) plane are expressed by the complex positions,
z
1
= x
1
+iy
1
and z
2
= x
2
+iy
2
, respectively. The complex potentials
corresponding to each vortex are
F
1
(z) =
k
1
2πi
log (z −z
1
), F
2
(z) =
k
2
2πi
log (z −z
2
). (7.45)
The total potential is given by F(z) = F
1
(z) +F
2
(z).
Each point vortex does not have its own proper velocity as shown
in Sec. 7.3.4, and the strengths k
1
and k
2
are invariants. Hence, the
vortex k
1
moves by the velocity induced by the vortex k
2
, and vice
versa. The complex velocity u −iv is given by the derivative dF/dz.
The complex velocity of vortex k
1
is the time derivative of ¯ z
1
(t) =
x
1
(t) −iy
1
(t) (complex conjugate of z
1
). Thus, equating d¯ z
1
/dt with
dF
2
/dz at z = z
1
, we have
d
dt
¯ z
1
=
dF
2
dz

ω
φ
r
2
πdxdr = 2πR
3
U.
7.6.2. Circular vortex ring
When the vortex-line is circular and concentrated on a circular core
of a small cross-section, the vortex is often called a circular vortex
ring. The stream function of a thin circular ring of radius R lying
in the plane x = 0 is obtained from (7.58) by setting ω
φ
(x

,
Hicks (1885) conﬁrmed Kelvin’s result for the vortex ring with uni-
form vorticity in the thin-core. In addition, he calculated the speed
of a ring of hollow thin-core, in which there is no vorticity within the
core and pressure is constant, as
U
h
=
γ
4πR
¸
log
8R
a
−
1
2

. (7.66)
7.7. Curved vortex ﬁlament
In this section, we consider three-dimensional problems. The ﬁrst
case is a curved vortex ﬁlament F of very small cross-section σ
with strength γ, embedded in an ideal incompressible ﬂuid in inﬁnite
space. We assume that the vorticity at a point y on the ﬁlament F
is ω(y) and zero at points not on the ﬁlament.
Considering a solenoidal velocity ﬁeld u(x) which satisﬁes
div u = 0, the vorticity is given by ω = curl u. This vanishes by
deﬁnition at all points except at y on F. The vortex ﬁlament F of
strength γ is expressed by a space curve of a small cross-section σ,
assumed to move about and change its shape. We denote its volume
element by dV = σ dl(y) where σ is the cross-section and dl(y) a
line element of the ﬁlament at y. Note that we have ωdV = γdl(y),
where γ = |ω| σ. The Biot–Savart law (7.5b) can represent velocity
u at x induced by a vorticity element, ωd
3
y = ωdV = γdl = γtds
at a point y(s). The velocity u(x) is expressed as
u(x) = −
γ
4π

F
(x −y(s)) ×t(s)
|x −y(s)|
3
ds, (7.67)
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166 Vortex motions
where s is an arc-length parameter along F, ds is an inﬁnitesimal
arc length and t unit tangent vector to F at y(s) (hence dl = tds).
We consider the velocity induced in the neighborhood of a point O
on the ﬁlament F. We deﬁne a local rectilinear frame K at O deter-
mined by three mutually-orthogonal vectors (t, n, b), where n and b
are unit vectors in the principal normal and binormal directions at
O (and t the unit tangent to F), as given in Appendix D.1. With the
point O as the origin of K, the position vector x of a point in the
plane normal to the ﬁlament F (i.e. perpendicular to t) at O can be
written as
x = yn +zb
(Fig. 7.7). We aim to ﬁnd the velocity u(x) obtained in the limit as x
approaches the origin O, i.e. r = (y
2
+z
2
)
1/2
→0.
In this limit it is found that the Biot–Savart integral (7.67) gives
u(x) =
γ
2π

y
r
2
b −
z
r
2
n

+
γ
4π
k
0

log
λ
r

b + (b.t.), (7.68)
where the ﬁrst two terms increase indeﬁnitely as r →0, and the term
(b.t.) denotes those that remain bounded. The ﬁrst term proportional
to γ/2π represents the circulatory motion about the vortex ﬁlament
F, counter-clockwise in the (n, b) plane, regarded as the right motion
so that this ﬁlament is said to be a vortex. However, there is another
term, i.e. the second term proportional to the curvature k
0
which is
not circulatory, but directed to b.
The usual method to resolve the unboundedness is to use a cut-
oﬀ. Namely, every vortex ﬁlament has a vortex core of ﬁnite size a,
and r should be bounded below at a value of order a. If r is replaced
y
z
r
y
(y, z)
x O t
n
n
O
b
Fig. 7.7. Vortex ﬁlament and local frame of reference (t, n, b).
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7.8. Filament equation (an integrable equation) 167
by a, the second term is
u
LI
=
γ
4π
k
0

log
λ
a

b, (7.69)
which is independent of y and z. This is interpreted such that the
vortex core moves rectilinearly with velocity u
LI
in the binormal
direction b.
The magnitude of velocity u
LI
is proportional to the local curva-
ture k
0
of the ﬁlament at O, and is called local induction. This term
vanishes with a rectilinear line-vortex because k
0
becomes zero. This
is consistent with the known property that a rectilinear vortex has
no self-induced velocity, noted in Sec. 7.3.4.
7.8. Filament equation (an integrable equation)
When we are interested in only the motion of a ﬁlament (without
seeing circulatory motion around it), the velocity would be given as
u
LI
(s), which can be expressed as
u
LI
(s) = ck(s) b(s) = ck(s) t(s) ×n(s), (7.70)
where c = (γ/4π) log(λ/a) is a constant independent of s. Rates
of change of the unit vectors (t, n, b) (with respect to s) along the
curve are described by the Frenet–Serret equation (D.4) in terms of
the curvature k(s) and torsion τ(s) of the ﬁlament.
3
A vortex ring is a vortex in the form of a circle (of radius R, say),
which translates with a constant speed in the direction of b. The
binormal vector b of the vortex ring is independent of the position
along the circle and perpendicular to the plane of circle (directed from
the side where the vortex-line looks clockwise to the side where it
looks counter-clockwise). The direction of b is the same as that of the
ﬂuid ﬂowing inside the circle. This is consistent with the expression
(7.70) since k = 1/R = const.
3
For a space curve x(s) = (x(s), y(s), z(s)), we have dx/ds = t(s), dt/ds =
k(s) n(s), dn/ds = −k(s) t − τ(s) b. See Appendix D.1 for more details.
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168 Vortex motions
It is found just above that the vortex ring in the rectilinear trans-
lational motion (with a constant speed) depicts a cylindrical surface
S
c
of circular cross-section in a three-dimensional Euclidean space
R
3
. The circular vortex ﬁlament f
c
coincides at every instant with
a geodesic line of the surface S
c
depicted by the vortex in space R
3
.
This is true for general vortex ﬁlaments f in motion under the law
given by Eq. (7.70), because the tangent plane to the surface S
f
to
be generated by vortex motion is formed by two orthogonal tangent
vectors t and u
LI
dt. Therefore, the normal N to the surface S
f
coin-
cides with the normal n to the curve f of the vortex ﬁlament. This
property is nothing but that f is a geodesic curve.
4
Suppose that we have an active space curve C: x(s, t), which
moves with velocity u
LI
. Namely, the velocity ∂
t
x at a station s
is given by the local value:
∂
t
x = u
∗
, u
∗
≡ ck(s) b(s),
where u
∗
is the local induction velocity. It can be shown that the
separation distance of two nearby particles on the curve, denoted by
∆s, is unchanged by this motion. In fact,
d
dt
∆s = (∆s∂
s
u
∗
) · t, (7.71)
where ∂
s
u
∗
= ck

(s) b + ckb

(s). From the Frenet–Serret equation
(D.4), it is readily seen that b · t = 0 and b

(s) · t = 0. Thus, it is
found that (d/dt)∆s = 0, i.e. the length element ∆s of the curve is
invariant during the motion, and we can take s as the Lagrangian
parameter of material points on C.
Since b = t ×n and in addition ∂
s
x = t and ∂
2
s
x = kn
(Appendix D.1), the local relation (7.70) for the curve x(s, t) is
given by
∂
t
x = ∂
s
x ×∂
2
s
x, (7.72)
where the time is rescaled so that the previous ct is written as t
here. This is termed the ﬁlament equation. In ﬂuid mechanics, the
4
A geodesic curve on a surface S is deﬁned by the curve connecting two given
points on S with a shortest (or an extremum) distance. See [Ka04, Sec. 2.6] for
its deﬁnition.
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7.9. Burgers vortex (a viscous vortex with swirl) 169
same equation is called the local induction equation (approximation).
5
Some experimental evidences are seen in [KT71].
It is remarkable that the local induction equation can be trans-
formed to the cubic-nonlinear Schr¨ odinger equation. Introducing a
complex function ψ(s, t) by
ψ(s, t) = k(s) exp
¸
i

s
τ(s

) ds

(called Hasimoto transformation [Has72]), where k and τ are the
curvature and torsion of the ﬁlament. The local induction equation
(7.72) is transformed to
∂
t
ψ = i

∂
2
s
ψ +
1
2
|ψ|
2
ψ

. (7.73)
As is well-known, this is one of the completely integrable systems,
called the nonlinear Schr¨odinger equation. This equation admits a
soliton solution, which propagates with a constant speed c along an
inﬁnitely long vortex ﬁlament as ψ(s, t) = k(s −ct) exp[iτ
0
s], where
k(s, t) = 2(τ
0
/α) sech (τ
0
/α)(s − ct) with τ = τ
0
=
1
2
c = const. and
α a constant.
7.9. Burgers vortex (a viscous vortex with swirl)
There is a mechanism for spontaneous formation of a vortex in a
certain ﬂow ﬁeld. In Sec. 10.4, we will see a mechanism of sponta-
neous enhancement of average magnitude of vorticity in turbulent
ﬂow ﬁelds.
The viscous vorticity equation without external force is written as
∂
t
ω + (v · grad)ω = (ω · grad)v +ν∇
2
ω, (7.74)
from (7.3). It is interesting to see that this equation has a solution
which shows concentration of vorticity without external means.
5
This equation was given by Da Rios (1906) and has been rediscovered several
times historically.
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170 Vortex motions
z
stream-line
vortex-line
Fig. 7.8. Burgers vortex under an external straining v
b
.
Suppose that the velocity ﬁeld consists of two components
(Fig. 7.8): v = v
b
+v
ω
. In the cylindrical frame of reference (r, θ, z),
the ﬁrst component v
b
is assumed to be axisymmetric and irrota-
tional, and represented as
v
b
= (−ar, 0, 2az), (7.75)
where a is a positive constant. This satisﬁes the solenoidal condition,
div v
b
= ∂
z
(v
b
)
z
+r
−1
∂
r
(r(v
b
)
r
) = 2a −2a = 0.
The component v
b
also has a velocity potential Φ. In fact,
v
b
= grad Φ, Φ = −
1
2
ar
2
+az
2
.
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7.9. Burgers vortex (a viscous vortex with swirl) 171
Hence, v
b
is clearly irrotational and regarded as a background strain-
ing ﬁeld which is acting on the second rotational component v
ω
,
assumed to be given as (0, v
θ
(r), 0). Its vorticity has only the axial z
component:
ω = ∇×v
ω
= (0, 0, ω(r, t)), ω = r
−1
∂
r
(rv
θ
).
The total velocity is given as
v = v
b
+v
ω
= (−ar, v
θ
(r), 2az). (7.76)
Then, the ﬁrst and last terms of the vorticity equation (7.74) have
only the z component. Similarly, the second term on the left and the
ﬁrst term on the right of (7.74) can be written respectively as
(v · grad)ω = (−ar∂
r
+v
θ
r
−1
∂
θ
+ 2az∂
z
)ω(r, t)
= (0, 0, −ar∂
r
ω(r, t)),
(ω · grad)v = ω∂
z
v = (0, 0, 2aω).
Hence both have only the z component as well. Thus, the vorticity
equation (7.74) reduces to the following single equation for the z
component:
∂
t
ω −ar∂
r
ω = 2aω +νr
−1
∂
r
(r∂
r
ω). (7.77)
In order to see the signiﬁcance of each term, let us consider ﬁrst
the equation formed only by the two terms on the left-hand side:
∂
t
ω −ar∂
r
ω = 0, neglecting the terms on the right. We immediately
obtain its general solution of the form ω = F(re
at
), where F is
an arbitrary function. One can easily show that this satisﬁes the
equation. At t = 0, the function ω takes the value F(r
0
) for r = r
0
.
At a later time t > 0, the same value is found at r = r
0
e
−at
< r
0
since a > 0. The same can be said for all values of r
0
∈ (0, ∞).
Therefore, the initial distribution of ω is convected inward at later
times and converges to r = 0 as t →∞. Thus, the term −ar∂
r
ω on
the left of (7.77) represents the eﬀect of an inward convection.
Next, the inﬂuence of the ﬁrst term on the right-hand side can
be seen by considering another truncated equation ∂
t
ω = 2aω. This
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172 Vortex motions
gives a solution ω(t) = ω(0)e
2at
, which represents an ampliﬁcation
of the vorticity. The term 2aω from (ω · grad)v represents vortex
stretching, as seen in Sec. 7.2.1. The last term νr
−1
∂
r
(r∂
r
ω) repre-
sents viscous diﬀusion.
The steady form of Eq. (7.77) is ar∂
r
ω+2aω+νr
−1
∂
r
(r∂
r
ω) = 0.
Its solution is readily found as [see Problem 7.9(i)].
ω
B
(r) = ω
B
(0) exp[−r
2
/l
2
B
], l
B
=

2ν/a. (7.78)
This steady distribution ω
B
(r) is called the Burgers vortex.
For an unsteady problem of the whole equation (7.77), a general
solution can be found in the form ω(r, t) = A
2
(t) W(σ, τ) in Problem
7.9(iii). As time tends to inﬁnity (t → ∞ for a > 0), this solution
tends to an asymptotic form of the Burgers vortex:
ω(t →∞) →ω
B
(r) =
Γ
πl
2
B
exp[−r
2
/l
2
B
], (7.79)
where Γ =

∞
0
Ω
0
(s) 2πs ds denotes the initial total vorticity [Ka84].
This asymptotic state is interpreted as follows. The vorticity is
swept to the center by a converging ﬂow v
b
. However, the eﬀect of
viscosity gives rise to an outward diﬀusion of ω. Thus, as a balance
of the two eﬀects, the vorticity approaches a stationary distribution
ω
B
(r) represented by the Gaussian function of (7.79). The parameter
l
B
represents a length scale of the ﬁnal form. The swirl velocity v
θ
(r)
around the Burgers vortex is given as
v
θ
(r) =
Γ
2πl
B
1
ˆ r

1 −e
−ˆ r
2

, ˆ r = r/l
B
. (7.80)
As a result of detailed analyses, evidences are increasing to show
that strong concentrated vortices observed in computer simulations
or experiments of turbulence have this kind of Burgers-like vortex.
This implies that in turbulence there exists a mechanism of sponta-
neous self-formation of Burgers vortices in the statistical sense.
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7.10. Problems 173
7.10. Problems
Problem 7.1 Vector potential A
Verify the relation curl v = ω by using the vector potential A deﬁned
by (7.5a). State in what cases the divA = 0 is satisﬁed.
Problem 7.2 Invariants of motion
Verify the conservation of the ﬁve integrals (7.9)–(7.11) of Impulse
P, Angular impulse L and Helicity H for the vorticity ω evolving
according to the vorticity equation (7.1).
Problem 7.3 Invariance of R
2
Verify that the integral (7.31) for R
2
is invariant during the vor-
tex motion, according to (7.27) for 2D motions of an incompressible
inviscid ﬂuid.
Problem 7.4 Vortex sheet
Suppose that a ﬂow is represented by (7.43) having a discontinuous
surface at y = 0. Show that its vorticity is given by the expression
(7.44).
Problem 7.5 Vortex ﬁlament
Derive the asymptotic expression (7.68) for the velocity u(x) from
the Biot–Savart integral (7.67)
Problem 7.6 Helical vortex
Show that the following rotating helical vortex x
h
= (x, y, z)(s, t)
satisﬁes Eq. (7.72):
x
h
= a(cos θ, sin θ, hks +λωt ), (7.81)
t
h
= ak(−sin θ, cos θ, h), θ = ks −ωt (7.82)
where a, k, h, ω, λ are constants and t
h
is its tangent.
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174 Vortex motions
Problem 7.7 Lamb’s transformation
[Lamb32, Sec. 162]
Consider a thin-cored vortex ring in a ﬁxed coordinate frame in which
velocity u(x, t) vanishes at inﬁnity. Suppose that the ﬂuid density is
constant and the vortex is in steady motion with a constant veloc-
ity U. Show the following relation between the kinetic energy K,
impulse P and velocity U:
K = 2 U· P+

u
∗
· (x ×ω) d
3
x, (7.83)
where u
∗
= u−U is the steady velocity ﬁeld in the frame where the
vortex ring is observed at rest.
Problem 7.8 Vortex ring
The kinetic energy K, impulse P and velocity U of a thin-cored
vortex ring of a ring radius R and a core radius a are given by
K =

dξ
1
dη
1
. (7.90)
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November 1, 2006 19:3 WSPC/Book -SPI-B364 “Elementary Fluid Mechanics” Trim Size for 9in x 6in ch08
Chapter 8
Geophysical ﬂows
Atmospheric motions and ocean currents are called
Geophysical Flows, which are signiﬁcantly inﬂuenced
by the rotation of the Earth and density stratiﬁcations
of the atmosphere and ocean. We consider the ﬂows in
a rotating system and the inﬂuence of density stratiﬁ-
cation on ﬂows compactly in this chapter.
1
This sub-
ject is becoming increasingly important in the age of
space science and giant computers.
8.1. Flows in a rotating frame
To study geophysical ﬂows, we have to consider ﬂuid motions in a
rotating frame such as a frame ﬁxed to the Earth, which is a non-
inertial frame. The equation of motion on such a noninertial frame
is described by introducing a centrifugal force and Coriolis force in
addition to the forces of the inertial system. It is assumed that a
frame of reference { is rotating with angular velocity Ω relative to
an inertial frame o ﬁxed to the space. The Navier–Stokes equation
in the inertial frame o is given by (4.10), which is reproduced here:
∂
t
v + (v ∇)v = −
1
ρ
∇p
m
+ν∆v,
1
ρ
∇p
m
=
1
ρ
∇p −g, (8.1)
1
For a more detailed account of the present subject (except Sec. 8.5), see for
example: [Hol04; Ach90, Sec. 8.5; Hou77] or [Tri77, Secs. 15, 16].
177
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178 Geophysical ﬂows
where p
m
is the modiﬁed pressure, deﬁned originally by (4.11) for
a constant ρ, and g is the acceleration due to gravity expressed by
−∇χ (χ: the gravity potential). The formula of p
m
can be extended
to variable density ρ. Assuming that the atmosphere is barotropic,
i.e. assuming ρ = ρ(p), one can write as ρ
−1
∇p = ∇Π by using Π
deﬁned below. Thus, we have
1
ρ
∇p
m
= grad P
m
, (8.2)
P
m
(x) := Π(p(x)) +χ(x), Π :=

p
p
0
dp
ρ(p)
, (8.3)
where p
0
is a reference pressure.
Suppose that the position of a ﬂuid particle is represented by x(t)
in the inertial frame o. Then, its velocity (Dx
S
/Dt)
S
= v
S
in the
frame o is represented as
v
S
=

Dx
S
Dt

S
=

Dx
R
Dt

R
+Ωx = v
R
+Ωx, (8.4)
where the subscript { refers to the rotating frame and x
S
= x
R
= x
since the coordinate origins are common. The expression (Dx
R
/Dt)
R
is the velocity relative to the rotating frame. The ﬁrst term v
R
denotes the velocity as it appears to an observer in the rotating
frame {, while the second term,
v
F
= Ωx, (8.5)
is a velocity due to the frame rotation which is an additional velocity
to an observer in the inertial frame o (Fig. 8.1) [see (1.30) for the
rotation velocity with an angular velocity
1
2
ω].
Analogously, the acceleration (Dv
S
/Dt)
S
in the inertial frame o
is given by taking time derivative of (8.4) in the frame o:
¸
D
Dt
v
S

z
0
g
∗
(z) dz, (8.12)
where g
0
= 9.807 m/s
−2
, the mean value at the surface.
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8.2. Geostrophic ﬂows 181
When there is no motion, i.e. v = 0, the above equation becomes
grad P = 0, which reduces to the hydrostatic equation:
dp
dz
= −ρ(z) g
∗
(z). (8.13)
8.2. Geostrophic ﬂows
From the previous section, it is seen that an essential diﬀerence
between the dynamics in a rotating frame from a nonrotating frame
is caused by the Coriolis term −2Ωv. We consider ﬂows that are
dominated by the action of Coriolis force, and suppose that the
eﬀect of Coriolis force is large compared with both of the eﬀects
of convection (v ∇)v and viscosity terms ν∇
2
v. This means
[Ωv[ [(v ∇)v[ and [Ωv[ [ν∇
2
v[.
Expressing these with the order of magnitude estimation in an anal-
ogous way to that carried out in Sec. 4.3, we obtain
ΩU U
2
/L and ΩU νU/L
2
,
or
R
o
:= U/ΩL <1 and E
k
:= ν/ΩL
2
<1. (8.14)
The dimensionless number R
o
= U/ΩL is known as the Rossby num-
ber, and E
k
= ν/ΩL
2
as the Ekman number, respectively.
When both the Rossby number and Ekman number are small and
the ﬂow is assumed steady, the equation of motion (8.11) reduces to
2Ωv = −grad P, (8.15)
where grad P is the resultant of grad p/ρ and grad χ
∗
= g
∗
from (8.9).
The velocity ﬁeld v ≡ v
g
(x) governed by this equation is called the
geostrophic ﬂow (Fig. 8.3). In the equipotential surface S
χ
deﬁned
by χ
∗
= const., we have from (8.15)
grad P

χ
∗
=const.
=
1
ρ
grad p = 2 v
g
Ω. (8.16)
Namely, grad P is equivalent to the pressure gradient in S
χ
.
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182 Geophysical ﬂows
v
Fig. 8.3. Geostrophic ﬂow.
An important property of the ﬂows on a rotating frame is now
disclosed. The Coriolis force 2Ω v is perpendicular to the veloc-
ity vector v. Hence the pressure gradient in the middle of (8.16)
is perpendicular to v
g
. Since the pressure gradient is perpendicular
to the constant pressure surface, the pressure is constant along the
stream-line. This is in marked contrast to the ﬂows in nonrotating
frame where ﬂow velocities are often parallel to the pressure gradient
(though not always so) and perpendicular to the constant pressure
surface.
This feature of geostrophic ﬂows is familiar in weather maps. The
weather maps are usually compiled from data of pressure obtained
at various observation stations. Iso-pressure contours are drawn on
a weather map, and these are also understood as meaning such lines
along which the wind is blowing, because the Earth’s rotation is a
dominating factor for the atmospheric wind. If it takes suﬃceint time
(i.e. one day or more) for the formation of a wind system such as low-
or high-pressures, the atmospheric motion is strongly inﬂuenced by
the Earth’s rotation.
4
In atmospheric boundary layers up to about 1 km or so from the
ground, the viscosity eﬀect cannot be neglected. In such a layer, the
wind direction would not necessarily be parallel to the iso-pressure
4
The wind should be observed at a suﬃcient height above the ground.
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8.3. Taylor–Proudman theorem 183
contours. This is known as Ekman boundary layer, which will be
considered later in Sec. 8.4.
8.3. Taylor–Proudman theorem
In a coordinate frame rotating with a constant angular velocity Ω,
steady geostrophic ﬂows are governed by Eq. (8.15) for an inviscid
ﬂuid. Setting the axis of frame of rotation as z axis, one can write
Ω = Ωk where Ω = [Ω[ and k is the unit vector in the direction of
z-axis of the Cartesian frame (x, y, z). Denoting the velocity satisfy-
ing (8.15) as v
g
= (u
g
, v
g
, w
g
), the (x, y, z) components of (8.15) are
written as
−2Ωv
g
= −∂
x
P, 2Ωu
g
= −∂
y
P, (8.17)
0 = −∂
z
P. (8.18)
Hence, the modiﬁed pressure P is constant along the z-direction (par-
allel to Ω)
5
and may be expressed as P = P(x, y), and it is found
that u and v are independent of z as well from (8.17). Thus, we have
∂
z
u = 0, ∂
z
v = 0, (8.19)
(the suﬃx g is omitted for brevity). In addition, the function P/(2Ω)
plays the role of a stream function for the “horizontal” component
(u, v), since v
h
:= (u, v, 0) = (1/2Ω)(−∂
y
P, ∂
x
P, 0) from (8.17).
6
This
means that the horizontal ﬂow is incompressible: ∂
x
u +∂
y
v = 0. All
of these (including the z-independent v
h
) are known as the Taylor–
Proudman theorem. The equation of continuity in the form of (3.8),
Dρ/Dt + div v = 0, reduces to
∂
z
w = −Dρ/Dt . (8.20)
If the ﬂuid is incompressible (i.e. Dρ/Dt = 0), we obtain ∂
z
w = 0.
5
grad P is the resultant of grad p/ρ and gradχ
∗
= g
∗
from (8.9).
6
If the plane z = 0 coincides with the equi-geopotential surface χ
∗
= const.,
then we have ρ (∂
x
, ∂
y
)P(x, y) = (∂
x
, ∂
y
)p, the pressure gradient in the horizontal
plane.
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184 Geophysical ﬂows
Vectorial representation of the horizontal component of v
g
is
given by
v
h
=
1
2Ω
k grad P, (8.21)
while the vertical z component w will be determined in the problem
below (Sec. 8.4). The z component of vorticity is given by
ω
g
≡ ∂
x
v −∂
y
u =
1
2Ω
(∂
2
x
+∂
2
y
)P. (8.22)
8.4. A model of dry cyclone (or anticyclone)
We now consider a geostrophic swirling ﬂow in a rotating frame and
its associated Ekman layer in which viscous eﬀect is important. We
are going to investigate a very simpliﬁed and instructive model of a
cyclone or an anticyclone (without moisture) in a rotating frame. In
particular, we are interested in a thin boundary layer of a geostrophic
ﬂow of a small viscosity over a ﬂat ground at which no-slip condition
applies. The boundary layer is called the Ekman layer. This is a
coupled system of a geostrophic ﬂow and an Ekman layer.
(a) An axisymmetric swirling ﬂow in a rotating frame
Suppose that we have a steady axisymmetric swirling ﬂow in a rotat-
ing frame, with its axis of rotation coinciding with the z axis. The ﬂow
external to the boundary layer is assumed to be a geostrophic ﬂow
over a horizontal ground at z = 0. Its vorticity ω
g
is given by (8.22)
and assumed to depend on the radial distance r only, i.e. ω
g
= ω(r)
where r = (x
2
+y
2
)
1/2
.
A typical example would be the vorticity of Gaussian function,
ω
G
= ω
0
exp[−r
2
/a
2
]
with a as a constant. For this solution, P = P(r) can be determined
from (8.22).
7
If the constant ω
0
is positive, the pressure is minimum
at r = 0 by (8.22) in each horizontal plane z = const., which cor-
responds to a low pressure (i.e. cyclone). This may be regarded to
7
By using (∂
2
x
+ ∂
2
y
)P = r
−1
D(rDP), where D ≡ d/dr from (D.11).
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8.4. A model of dry cyclone (or anticyclone) 185
be a very simpliﬁed model of tropical cyclone (without moisture). If
ω
0
< 0, the pressure is maximum at r = 0, corresponding to a high
pressure (anticyclone).
In the polar coordinate (r, θ) of the horizontal (x, y)-plane where
θ = arctan(y/x), the horizontal velocity v
h
is given by the azimuthal
component V
θ
(r) with vanishing radial componet V
r
:
V
θ
(r) =
Γ
2π
1
r
(1 −e
−r
2
/a
2
), V
r
= 0, (8.23)
(see (7.80)), where Γ = πa
2
ω
0
. In fact, we have ω = r
−1
∂
r
(rV
θ)
= ω
G
by (D.15). Typical length scale of this swirling ﬂow (8.23) is a, while
typical magnitude of this ﬂow is given by U
s
≡ Γ/2πa.
Cartesian components u
g
and v
g
of v
h
are given by
u
g
(x, y) = −V
θ
(r)sin θ, v
g
(x, y) = V
θ
(r)cos θ (8.24)
(x = r cos θ, y = r sin θ). Substituting these into (8.17) and assuming
P = P(r), we obtain the pressure gradient, dP(r)/dr = 2ΩV
θ
(r).
(b) Ekman boundary layer
The signiﬁcance of the Ekman boundary layer is seen clearly and
is instructive in this axisymmetric model. Motion of the main body
of ﬂuid away from the boundary is the axisymmteric ﬂow of swirl
considered in (a). In other words, the horizontal (x, y)-plane at z = 0
is rotating with the angular velocity Ω with respect to the ﬁxed
inertial frame, whereas the main body of ﬂuid away from the plane
z = 0 is the axisymmetric swirling ﬂow of the velocity v
h
= (0, V
θ
(r))
in addition to the frame rotation. Namely, the z component of total
vorticity is given by 2Ω + ω
G
against the inertial frame since the
uniform rotation of the frame has the vorticity 2Ω.
The ﬂuid viscosity ν is assumed to be very small, so that the
boundary layer adjacent to the wall is very thin in a relative sense.
Then, the governing equation is given by
2Ωv = −grad P +ν∇
2
v, (8.25)
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186 Geophysical ﬂows
from (8.11),
8
instead of (8.15) for inviscid geostrophic ﬂows. We
assume that the Reynolds number R
e
is much larger than unity,
R
e
:= U
s
a/ν 1.
Now we consider the boundary layer adjacent to the ground z =
0. Just like the boundary layer of nonrotating system investigated
in Sec. 4.5, it is assumed that variations (derivatives) of velocity
v = (u, v, w) with z are much larger than those with x or y. It is
found that Eq. (8.25) reduces to
−2Ωv = −∂
x
P +ν ∂
2
z
u, (8.26)
2Ωu = −∂
y
P +ν ∂
2
z
v, (8.27)
0 = −∂
z
P +ν ∂
2
z
w. (8.28)
The continuity equation (3.8) is
∂
x
u +∂
y
v +∂
z
w + Dρ/Dt = 0. (8.29)
From (8.29), we deduce that w is much smaller than the horizontal
components, and estimate as [w[/U
s
= O(δ/a) <1 according to the
usual argument of boundary layer theory (see Problem 4.4(i) and its
solution), where δ is a representative scale of the boundary layer.
By applying the same argument of Problem 4.4(i) to (8.28), we
obtain
[∆P[
U
2
s
= O(R
−1
e
), R
e
= U
s
a/ν 1.
Namely, variation ∆P across the boundary layer normalized by U
2
s
is very small and may be neglected. Therefore, the pressure in the
boundary layer is imposed by that of the external ﬂow. The ﬂow
external to the boundary layer is the swirling ﬂow (8.23) whose P
is essentially a function of x and y only (independent of z). The
derivatives ∂
x
P and ∂
y
P are given by (8.17).
Eliminating ∂
x
P by using (8.17), Eq. (8.26) is reduced to the form
−2Ω(v −v
g
) = ν∂
2
z
u. The right-hand side can be written as ν∂
2
z
(u −
u
g
) since u
g
is independent of z. Therefore, we have −2Ω(v − v
g
) =
8
In the axisymmetric swirling ﬂow under consideration, the term (v · ∇)v =
V
θ
∂
θ
v + w∂
z
v on the left of (8.11) vanishes (for incompressible ﬂows).
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8.4. A model of dry cyclone (or anticyclone) 187
ν ∂
2
z
(u−u
g
). Similarly, Eq. (8.27) is reduced to the form 2Ω(u−u
g
) =
ν ∂
2
z
(v −v
g
) by eliminating ∂
y
P. Thus, we obtain
−2Ω(v −v
g
) = ν ∂
2
z
(u −u
g
), (8.30)
2Ω(u −u
g
) = ν ∂
2
z
(v −v
g
). (8.31)
This can be integrated immediately. In fact, multiplying (8.31) by
the imaginary unit i =
√
−1 and adding it to Eq. (8.30), we obtain
2Ωi X = ν∂
2
z
X, (8.32)
X := (u −u
g
) +i(v −v
g
). (8.33)
Solving (8.32), we immediately obtain a general solution:
X = Ae
−(1+i)ζ
+ Be
(1+i)ζ
, ζ ≡ z/δ, (8.34)
δ =

ν
4Ω
ω
G
(x, y). (8.37)
It is remarkable to ﬁnd that there is a nonzero vertical ﬂow w
E
at
the outer edge of the Ekman layer, which is positive or negative
according as the sign of ω
0
. Namely, in the case of low pressure (or
like a cyclone), it is an upward ﬂow (Fig. 8.4). This is connected with
the inward ﬂow within the boundary layer, and termed as Ekman
pumping. The total inﬂux F
in
at a position r within the Ekman layer
across a cylindrical surface of radius r is given by the integral of
inﬂux,

ν
Ω
πa
2
ω
0
(1 −exp[−r
2
/a
2
]).
10
R
∞
0
e
−ζ
(sinζ) dζ =
1
2
.
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8.5. A model of dry cyclone (or anticyclone) 189
ω
Fig. 8.4. Schematic sketch of a dry cyclone with Ekman boundary layer on a
system rotating with Ω. A geostrophic swirling ﬂow of velocity V
θ
(r) with an
axial vorticity ω
G
(r) of Gaussian form induces upward pumping ﬂow w
E
(r) at
the edge of the Ekman layer.
This kind of structure of low pressure is often observed in tropical
cyclones, although this model is dry and very simpliﬁed. In the case
of high, w
E
is downward and connected with the outward ﬂow within
the Ekman layer.
Finally, a remark must be made on real atmospheric conditions.
Observations indicate that the wind velocity in the atmospheric
boundary layer approaches its upper geostrophic value at about
1 km above the ground. If we let 5δ = 1000 m and Ω = 10
−4
(≈ 2π/(246060)) s
−1
for the earth spin, the deﬁnition δ =

ν/Ω
implies the value of ν about 4 10
4
cm
2
s
−1
[Hol04]. This is much
larger than the molecular viscosity.
The traditional approach to this problem is to assume that turbu-
lent eddies act in a manner analogous to molecular diﬀusion, so that
the momentum ﬂux is dominated by turbulent action, and the vis-
cosity terms in (8.26)–(8.28) are replaced by ν
turb
∂
2
z
v, where ν
turb
is
the turbulent viscosity which is found as ν
turb
≈ 410
4
cm
2
s
−1
in the
above estimate. Later in Sec. 10.1, we will consider Reynolds stress
R
ij
= ρ 'u

i
u

j
`. The above is equivalent to ∂
j
R
ij
= −ρν
turb
∂
2
z
'u
i
`. In
this case, average velocity ﬁeld must be considered.
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190 Geophysical ﬂows
8.5. Rossby waves
Rossby wave is another aspect of ﬂows resulting from the rotation of
a system, which is responsible for a certain wavy motion of the atmo-
sphere. Waves of global scales observed on the Earth (or planets) are
known as planetary waves (Fig. 8.5). Rossby wave is one such wave
which occurs owing to the variation of Coriolis parameter (f deﬁned
below) with the latitude φ. For the global scales or synoptic scales L
(typically 1000 km or larger in horizontal dimension which are very
much larger than the vertical scale of atmosphere of about 10 km),
Fig. 8.5. Planetary waves, exhibited by the geopotential height z
g
(m) of a
constant-pressure surface (of 500 hPa corresponding to about 5000 m) in the
northern hemisphere φ > 15
◦
on 31 Oct. 2005, plotted with the NCEP/NCAR
reanalysis data, by the courtesy of Dr. T. Enomoto (The Earth Simulator Center,
Japan).
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8.5. Rossby waves 191
vertical velocities are very much smaller than horizontal velocities
(see the next section), so that the vertical component of velocity v
can be neglected in the equation of motion. Such a motion is termed
as quasi-horizontal.
The viscous term ν∇
2
v is neglected. The equation of motion
(8.11) reduces to
D
t
v + 2Ωv = −grad P, D
t
v ≡ ∂
t
v +v ∇v. (8.38)
A convenient frame of reference at a point O on the earth’s sur-
face is given by the cartesian system of x directed towards the
east, y towards the north and z vertically upwards, where the lat-
itude of the origin O is denoted by φ (Fig. 8.6). Using i, j, k for
unit vectors along respective axis and writing v = (u, v, w) and
Ω = Ω(0, cos φ, sin φ), we have
2Ωv = 2Ω(w cos φ −v sin φ)i + 2Ωu sin φj −2Ωu cos φk.
A simplest Rossby wave solution is obtained for an atmosphere of
constant density ρ under the assumption of no vertical motion w = 0.
equator
O
x
y
N
Fig. 8.6. β-plane.
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192 Geophysical ﬂows
The equations for a horizontal motion are, from (8.38) and (3.9),
D
t
u −fv = −∂
x
P, (8.39)
D
t
v + fu = −∂
y
P, (8.40)
∂
x
u +∂
y
v = 0, (8.41)
f = 2Ω sinφ : (Coriolis parameter), (8.42)
and D
t
= ∂
t
+u∂
x
+v∂
y
by neglecting terms of O(L/R) because the
reference frame is a curvilinear system, where L is the scale of ﬂow
and R the radius of the earth (assumed a sphere).
Operating ∂
y
on (8.39) and ∂
x
on (8.40), and subtracting, we
obtain
11
D
t
ζ +v∂
y
f = 0, (8.43)
(see Problem 8.2), where ζ = ∂
x
v − ∂
y
u is the z component of the
vorticity curl v, and the termf (∂
x
u+∂
y
v) was omitted due to (8.41).
Since f (= 2Ω sin φ) depends only on y = a(φ − φ
0
) (where φ(O) =
φ
0
), the above equation can be written in the form,
D
t
[ζ +f] = 0. (8.44)
The quantity ζ +f is regarded as the z component of absolute vortic-
ity, for ζ is the z component of vorticity which is twice the angular
velocity of local ﬂuid rotation and f is twice the angular velocity
of frame rotation around the vertical axis. Therefore Eq. (8.44) is
interpreted as the conservation of absolute vorticity ζ +f.
In order to ﬁnd an explicit solution of (8.44), we assume a linear
variation of f with respect to y, i.e. f = f
0
+ βy where β is a con-
stant as well as f
0
. This is known as the β-plane approximation.
12
An unperturbed state is assumed to be a uniform zonal ﬂow (¯ u, 0).
Writing the perturbed state as u = ¯ u + u

s
= dρ
s
/dz, and W denotes the scale of vertical velocity
w. In the Boussinesq approximation, the density variation is taken
into consideration only in the buoyancy force and the density on the
left-hand side of (8.52) is given by a representative density ρ
0
. Then
Eq. (8.52) becomes
ρ
0
ω v +ρ
0
grad

s
[
=
1
R
i
= (F
r
)
2
, (8.57)
where R
i
is called the Richardson number, and F
r
the Froud
number.
14
When the Froud number is small (or equivalently, the Richardson
number is large), the vertical motion is much weaker than the hori-
zontal motion. That is the case for most geophysical ﬂows in which
the length scale L is of the order of earth itself, or so. Actually,
the Richardson number is very large in such ﬂows and horizontal
motions are predominating. The jet-stream in the stratosphere is a
typical example of such a horizontal ﬂow.
In a laboratory experiment too, such a suppression of vertical
motion of a stratiﬁed ﬂow can be observed in the visualization exper-
iment of Fig. 8.7.
8.7. Global motions by the Earth Simulator
The Earth Simulator (ES)
15
constructed in Japan as a national
project (proposed in 1997) was the largest and fastest computer
in the world as of March 2003. The computer started its opera-
tion in February 2002. The project aims for computer simulations
of global motions of atmosphere and ocean that previous computers
were unable to accomplish, as well as simulations of slow plastic vis-
cous motions of the earth’s interior, and also predictions of future
climate and environment of the earth. This is a gigantic computer
consisting of 5,120 super-computer units. The performance of the
computing speed was reported as 36Tﬂops, i.e. 36 10
12
operations
per second.
14
The Froud number is deﬁned by F
r
= U/
√
gL in hydraulics, where the density
changes discontinuously at the free water surface and L|ρ

s
| is replaced by ρ
0
.
15
The Earth Simulator Center is located at JAMSTEC (Japan Agency for Marine-
Earth Science and Technology), Kanazawa-ku, Yokohama, Japan, 236-0001. Web-
site: http://www.es.jamstec.go.jp/esc/eng/ESC/index.html; Journal of the Earth
Simulator can be viewed at this site.
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8.7. Global motions by the Earth Simulator 197
Fig. 8.7. Suppression of vertical motion in a laboratory experiment: Flow of a
water (from left to right) stratiﬁed vertically of density ρ
s
(z) (with a stable linear
salinity component) around a horizontal circular cylinder (on the right) of diam-
eter d = 1.0 cm and U ≈ 1.3 cm/s at Re = Ud/ν ≈ 113. Froude number is F
r
=
(a) ∞ (no stratiﬁcation), (b) 0.2, and (c) 0.1 (with L = 10d). The photograph is
provided through the courtesy of Prof. H. Honji (Kyushu University) [Hon88].
In numerical weather predictions carried out by previous ordinary
super-computers, the horizontal mesh scale was of the order of 100 km
or so, while by the Earth Simulator it is 10 km or less. This was made
possible by the improvement in performance by 10
3
times of previous
computers. In future, it aims for 3 km or less horizontal resolution,
in order to improve predictions of local weather considerably.
In the ﬁrst three years, experimental computer simulations have
been carried out by new computation codes in order to test the
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198 Geophysical ﬂows
performance of the giant computer ES, and some innovative results
have been obtained already. The next two subsections are brief
accounts of the results of global atmospheric motions and ocean cir-
culations, reported in the Journal of the Earth Simulator (see the
footnote).
8.7.1. Simulation of global atmospheric motion
by AFES code
Experimental test simulations of global atmospheric motions were
carried out by the AFES code (Atmospheric General Circulation
Model for the Earth Simulator), which is a new computation code.
This enabled global coverage of both the large-scale general circula-
tions and meso-scale phenomena, and their interaction [AFES04].
The ﬁrst experimental computation was a long-term simulation of
twelve years, with intermediate resolutions. The initial condition was
the data set of January 1, 1979, with the boundary conditions of daily
ﬁelds of sea-surface temperature, sea-ice cover and surface topogra-
phy. Next, using the resulting data, additional short-term simulations
were performed with the ﬁnest meshes of about 10 km horizontal scale
and 20 to 500 m altitude resolution (with ﬁner meshes in the atmo-
spheric boundary layer), in order to ﬁnd detailed local evolutions
such as (a) Winter cyclones over Japan, (b) Typhoon genesis over
the tropical western Paciﬁc, and (c) Cyclone-genesis in the south-
ern Indian Ocean. It has been found that the interactions among
global atmospheric motions and regional meso-scales are simulated
in a fairly realistic manner.
8.7.2. Simulation of global ocean circulation
by OFES code
Experimental test simulations of global ocean circulations were car-
ried out by the OFES code, which is an optimized code of MOM3
(Oceanic General Circulation Model) developed by GFDL (USA),
adapted to the Earth Simulator. The OFES enabled a ﬁfty-year
eddy-resolving simulation of the world ocean general circulation
within one month [OFES04].
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8.7. Global motions by the Earth Simulator 199
The initial conditions for the experimental integration of 50 years
are the annual mean sea temperature and salinity ﬁelds without
motion of sea water. Hence, this is called a spin-up problem, and
also called a climatological 50-year Integration. The computation
domain was the area from 75
◦
S to 75
◦
N (except the Arctic ocean)
with the horizontal grid spacing 1/10
◦
(about 10 km) and 54 vertical
levels. The computation used the monthly mean wind stresses aver-
aged from 1950 to 1999 (hence twelve data sets) taking account of
fresh water inﬂux from both precipitation and rivers.
In the spin-up computation, ﬁve active regions have been identi-
ﬁed (Fig. 8.8), where the sea surface height variability is signiﬁcantly
large: Kuroshio current in the west of northern Paciﬁc, Gulf stream in
the west of northern Atlantic, the three currents in the south western
Indian Ocean, south western Paciﬁc, south western Atlantic. In addi-
tion, the sea surface height of the Agulhas rings was clearly and real-
istically visualized. These results demonstrate a promising capability
of OFES, and the 50-year spin-up run represent realistic features of
the world ocean both in the mean ﬁelds and eddy activities.
Following the spin-up run, a hindcast run was carried out to study
various ocean phenomena and compare with the past observations.
Fig. 8.8. Five active regions visualized by OFES simulation [OFES04].
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200 Geophysical ﬂows
Among others, it is remarkable that the modes of El Nino and Indian
Ocean Dipole have been found and are in good agreement with the
observations. In view of the fact that tropical oceans play major
roles in the variability of world climate, this ﬁnding is encouraging
for future investigations.
These results show that hyper-resolution computations can be
powerful tools for investigating not only the large-scale global circu-
lations, but also the meso-scale phenomena such as cyclone-genesis,
eddies and instability, and thirdly the interaction between global-
scales and meso-scales, spatially and temporally.
Owing to the Earth Simulator, the study of global atmosphere
and ocean has become a science in the sense that experimental tests
can be done and quantitative comparison with observations is made
possible. This is in contrast with the analyses in the previous sections:
Sec. 8.2 for geostrophic ﬂows, Sec. 8.5 for Rossby waves and Sec. 8.6
for stratiﬁed ﬂows. They are all local analyses or approximate model
analyses, not a global analysis.
8.8. Problems
Problem 8.1 Thermal wind and jet stream
Horizontal pressure gradient arises in the atmosphere owing to tem-
perature variation which is related to the horizontal density gradient.
This is responsible to the thermal wind in a rotating system. Global
distribution of average atmospheric temperature
¯
T(x, y, z) generates
a thermal wind, called the jet stream (i.e. westerly).
Assuming 2Ω = f k (f = 2Ωsin φ) in (8.15) with k the unit
vertical vector in the local β-plane approximation (of Sec. 8.5), we
have the horizontal component v
h
(projection to the (x, y)-plane)
of the geostrophic velocity given by (8.21): v
h
= (1/f)k grad P.
Answer the following questions.
(i) Suppose that z = z
p
(x, y) represents a surface of the altitude
of constant pressure (p = const.). Using the hydrostatic relation
(8.13), show the following equation,
ρg
∗
grad z
p
(x, y) = grad p(x, y; χ
∗
= const.). (8.58)
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8.8. Problems 201
[Hint: On the surface z = z
p
(x, y), dp[
y=const
= ∂
x
p dx +
∂
z
p dz
p
= 0.]
Thus, the pressure gradient in the horizontal surface χ
∗
= const.
is related to the inclination of the surface z = z
p
(x, y). The global
mean pressure in the northern hemisphere is higher toward the south
(i.e. toward the equator), because the temperature is higher in the
south. In the horizontal plane, we obtain ρ grad P = grad p, since
χ
∗
= const. (see the footnotes to Sec. 8.3). Therefore, the horizontal
component of geostrophic velocity v
h
(proportional to k grad P)
is toward the east (i.e. westerly). This will be considered in terms of
the temperature in the next questions.
(ii) Suppose that we consider two p-constant surfaces p
0
and
p
∗
(< p
0
) and the layer in-between, by using the equation of state
p = Rρ T for an ideal gas (R = R
∗
/µ
m
, R
∗
: the gas constant; see
the footnote to Sec. 1.2)). Show the following relation (8.59):
grad P[
p=p
∗
−grad P[
p=p
0
= grad χ
∗
(p
∗
) −grad χ
∗
(p
0
)
= RP(p
∗
) grad
¯
T, (8.59)
where χ(p
∗
) −χ(p
0
) =

z(p
∗
)
z(p
0
)
g
∗
(z) dz = −

p
∗
p
0
dp
ρ
(8.60)
= RΠ(p
∗
)
¯
T(x, y), (8.61)
Π(p
∗
) =

p
0
p
∗
dp/p,
¯
T(x, y) =
1
z −z
0

z
z
0
T(x, y, z) dz.
(
¯
T(x, y) is the vertical average of temperature). Thus, we obtain
the equation for the thermal wind:
v
h
(p
∗
) −v
h
(p
0
) =
1
f
RP(p
∗
)k grad
¯
T. (8.62)
(iii) Suppose that the temperature
¯
T is higher toward the south and
lower toward the north and uniform in the east-west direction.
This occurs often in the mid-latitudes of the northern hemi-
sphere and referred to as baroclinic [Hou77]. Which direction
the geostrophic velocity v
h
(p
∗
) is directed if v
h
(p
0
) = 0?
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Chapter 9
Instability and chaos
Not every solution of equations of motion can actu-
ally occur in Nature, even if it is exact. Those which
do actually occur not only must obey the equations of
ﬂuid dynamics, but must be also stable (Landau and
Lifshitz [LL87, Sec. 26]).
In order that a speciﬁc steady state S can be observed in nature,
the state must be stable. In other words, when some external pertur-
bation happens to disturb a physical system, its original state must
be recovered, i.e. the perturbation superposed on the basic state S
must decay with time. In nature there always exists a source of dis-
turbance. One of the basic observations in physics is as follows: if a
certain macroscopic physical state repeatedly occurs in nature, then
it is highly possible that the state is characterized by a certain type
of stability.
If a small perturbation grows with time (exponentially in most
cases), then the original state S is said to be unstable. In such a
case, the state would have little chance to be observed in nature. If
the perturbation neither grows nor decays, but stays at the initial
perturbed level, then it is said to be neutrally stable.
When the initial state is unstable and the amplitude of pertur-
bation grows, then a nonlinear mechanism which was ineﬀective at
small amplitudes makes its appearance in due course of time. This
nonlinearity often suppresses further exponential growth of pertur-
bation (though not always so) and the amplitude tends to a new
203
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204 Instability and chaos
ﬁnite value. The new state thus established might be a steady state,
a periodic state, or irregularly ﬂuctuating state. The last one is often
said to be turbulent. If the initial state is a steady smooth ﬂow, then
it is said to be laminar.
9.1. Linear stability theory
Stability analysis of a steady state (called a basic state) is carried
out in the following way. Basic state whose stability is to be studied
is often a laminar ﬂow. Various laminar ﬂows of viscous ﬂuids are
presented in Chapter 4: boundary layer ﬂows, parallel shear ﬂows,
rotating ﬂows and ﬂows around a solid body, etc.
Suppose that the basic state is described by a steady velocity ﬁeld
v
0
(x) and an inﬁnitesimal perturbation velocity v
1
(x, t) is superim-
posed on it. Then the total velocity is v = v
0
+v
1
. It is assumed that
the velocity satisﬁes the divergence-free condition and the density ρ
0
is a constant. Then the equation of motion is given by
∂
t
v + (v · ∇)v = −(1/ρ
0
) ∇p +ν∆v +f , (9.1)
(see (4.8)), supplemented with div v = 0. When we consider a sta-
bility problem of an ideal ﬂuid, the viscosity ν is set to be zero.
Let us write the velocity, pressure and force as
v = v
0
+v
1
, p = p
0
+p
1
, f = f
0
+f
1
.
Suppose that the basic steady parts v
0
, p
0
, f
0
are given. The equation
the basic state satisﬁes is
(v
0
· ∇)v
0
= −(1/ρ
0
) ∇p
0
+ν∆v
0
+f
0
, (9.2)
and div v
0
= 0. Subtracting this from (9.1), we obtain an equa-
tion for the perturbations. Since the perturbations are assumed to
be inﬁnitesimal, only linear terms with respect to perturbations
v
1
, p
1
, f
1
are retained in the equation, and higher-order terms are
omitted. Thus we obtain a linearized equation of motion:
∂
t
v
1
+(v
0
· ∇)v
1
+(v
1
· ∇)v
0
= −(1/ρ
0
) ∇p
1
+ν∆v
1
+f
1
. (9.3)
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9.1. Linear stability theory 205
One can carry out the so-called the normal mode analysis, since the
coeﬃcients of (9.3) to the perturbations v
1
, p
1
and f
1
are all time-
independent, so that a time factor of the ﬂuctuating components
is represented by an exponential function, say e
αt
. In such a case,
the ﬂuctuating components v
1
, p
1
and f
1
can be represented by the
product of e
αt
and functions of the spatial variable x. For example,
the velocity v
1
is represented as
v
1
(x, t, α) = e
(α
r
+iα
i
)t
u(x, α), (9.4)
where α = α
r
+ iα
i
is the complex frequency. We have analogous
expressions for p
1
and f
1
. (This corresponds to a Fourier analysis.)
In general, the solutions satisfying the above equation (9.3) and
boundary conditions (to be speciﬁed in each problem) are determined
for particular eigenvalues of α, which may be discrete or continuous,
and in addition, complex in general.
1
Correspondingly, the eigen-
solutions v
1
(x, t, α), p
1
(x, t, α) and f
1
(x, t, α) are complex. A gen-
eral solution is given by a linear superposition of those particular
solutions.
If the real part α
r
of eigenvalue α is negative, then the ﬂuctu-
ation decays exponentially with time, whereas if it is positive, the
ﬂuctuation grows exponentially with time. Thus it is summarized as
stable if α
r
< 0; unstable if α
r
> 0.
The imaginary part α
i
of the eigenvalue gives the oscillation fre-
quency of the ﬂuctuation as seen from (9.4). Suppose that we con-
sider a transition sequence in which the eigenvalue α
r
changes from
negative to positive values when α
i
= 0. In the beginning, the ﬂuc-
tuating state with α
r
< 0 decays to a steady state. Once α crosses
the state α
r
= 0, a growing mode v
1
with a frequency α
i
appears.
This is understood as the Hopf bifurcation.
The above analysis is called linear stability theory. In sub-
sequent sections, we consider the instability and chaos by selecting
typical example problems, i.e. the Kelvin–Helmholtz instability, sta-
bility of parallel shear ﬂows and thermal convection.
1
Very often, solutions are determined only for complex values of α.
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206 Instability and chaos
9.2. Kelvin–Helmholtz instability
One of the most well-known instabilities in ﬂuid mechanics is the
instability of a vortex sheet which is a surface of discontinuity of
tangential velocity (Problem 7.4). The Kelvin–Helmholtz theorem
states that the vortex sheet is unstable in inviscid ﬂuids. Another
well-known instability is the Rayleigh–Taylor instability, which will
be considered in Problem 9.1.
Let us investigate the motion of the surface of discontinuity
located at y = 0 in the unperturbed state [Fig. 4.5(a)]. It is assumed
that the velocity of the basic state in (x, y)-plane is given by

1
2
U, 0

for y < 0;

−
1
2
U, 0

for y > 0. (9.5)
9.2.1. Linearization
Suppose that due to a perturbation the surface of discontinuity is
deformed and described by
y = ζ(x, t). (9.6)
Both above and below the surface, it is assumed that the ﬂow is
irrotational and the velocity potential is expressed as
Φ =

= 0. (9.13)
Thus, it is found (for k = 0) that
σ = ±
1
2
kU. (9.14)
2
This is equivalent to the Fourier mode analysis with respect to the x space. The
mode e
ikx
is orthogonal to another mode e
ik

x
for k

= k. Arbitrary perturbation
can be formed by integration with k, e.g.
R
f(k, y)e
ikx
dk = φ(x, y).
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9.3. Stability of parallel shear ﬂows 209
The surface form (for a real A) is given by the real part of (9.11),
ζ = Ae
σt
cos kx (9.15)
(the imaginary part is a solution as well).
The solution (9.15) with (9.14) has both modes of growing and
decaying with time t corresponding to σ =
1
2
kU and σ = −
1
2
kU,
respectively. A general solution is given by a linear combination of
two such modes. Therefore, it grows with t in general for any k(> 0).
Thus, the vortex sheet under consideration is unstable, and waves of
sinusoidal form grow with time along the sheet. It is remarkable that
the basic state (9.5) is unstable for any wavenumber k(>0). Finally,
it develops into a sequential array of eddies [Fig. 9.1(b)]. This is
called the Kelvin–Helmholtz instability.
A similar analysis can be made for the Rayleigh–Taylor problem
of a heavier ﬂuid placed over a lighter ﬂuid in a constant gravitational
ﬁeld (see Problem 9.1).
9.3. Stability of parallel shear ﬂows
One of the well-studied problems of ﬂuid mechanics is the stability
of parallel shear ﬂows [DR81]. A typical example is the stability of a
two-dimensional parallel ﬂow of velocity in (x, y)-plane,
v
0
= (U(y), 0), −b < y < b, (9.16)
in a channel between two parallel plane walls at y = −b, b. The ﬂuid
is assumed to be incompressible, either inviscid or viscous. This type
of ﬂow, i.e. a parallel shear ﬂow, is a typical laminar ﬂow. Some
steady solutions v
0
(y) of this type are presented in Sec. 4.6.
The ﬂow is governed by the two-dimensional Navier–Stokes equa-
tions, (4.29)–(4.31). Suppose that a perturbation v
1
= (u
1
, v
1
) is
superimposed on the steady ﬂow v
0
and the total velocity is writ-
ten as v = v
0
+ v
1
. The linearized equation for the perturbation is
given by (9.3) with f
1
= 0. In reality, perturbation must be investi-
gated as a three-dimensional problem. However, it can be shown that
a two-dimensional perturbation is most unstable (Squire’s theorem,
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210 Instability and chaos
Problem 9.4). Hence, we restrict ourselves to a two-dimensional prob-
lem here.
In the two-dimensional problem, one can introduce a stream func-
tion ψ(x, y, t) for an incompressible ﬂow (see Appendix B.2) by
u
1
= ∂
y
ψ, v
1
= −∂
x
ψ. (9.17)
9.3.1. Inviscid ﬂows (ν = 0)
Setting ν = 0, and v
0
= (U(y), 0), v
1
= (u
1
, v
1
), f
1
= 0 in the
perturbation equation (9.3), its x and y components are reduced to
∂
t
u
1
+U(y)∂
x
u
1
+v
1
U

(y)∂
x
ψ = 0. (9.20)
The boundary condition is
v
1
= −∂
x
ψ = 0, at y = −b, b. (9.21)
In view of the property that the coeﬃcients of Eq. (9.20) are inde-
pendent of x, we can take the same normal mode e
ikx
as before with
respect to the x coordinate (with time factor specially chosen as
e
−ikct
), assuming
ψ(x, y, t) = φ(y)e
ik(x−ct)
. (9.22)
We substitute this into (9.20) and carry out the replacements ∂
x
=
ik and ∂
t
= −ikc. Dropping the common exponential factor and
dividing by ik, we obtain
(U −c) (D
2
−k
2
)φ −U

φ = 0, (9.23)
where D ≡ d/dy. This is known as the Rayleigh’s equation (Rayleigh
1880). The boundary condition (9.21) reduces to
φ = 0, at y = −b, b. (9.24)
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9.3. Stability of parallel shear ﬂows 211
Equations (9.23) and (9.24) constitute an eigenvalue problem. Given
a real wavenumber k, this boundary-value problem would be solved
with an eigenvalue value c
∗
. In general, this c
∗
is a complex number
and written as c
∗
= c
r
+ ic
i
. The imaginary part determines the
growth rate kc
i
of the perturbation, since we have from (9.22)
ψ(x, y, t) = φ(y) e
kc
i
t

cos k(x −c
r
t) +i sin k(x −c
r
t)

. (9.25)
Thus the basic state of parallel ﬂow U(y) is said to be
stable if c
i
< 0, unstable if c
i
> 0.
The perturbation is a traveling wave of phase velocity c
r
.
For the purpose of stating the Rayleigh’s inﬂexion-point theorem
for inviscid parallel shear ﬂow, let us rewrite Eq. (9.23) in the fol-
lowing way:
φ

−k
2
φ −
U

U −c
φ = 0. (9.26)
If c
i
= 0, the factor U − c = U − c
r
− ic
i
in the denominator of the
third term never becomes zero.
Rayleigh’s inﬂexion-point theorem: A neccesary condition for
instability is that the basic velocity proﬁle U(y) should have an
inﬂexion point. [Fig. 9.2. See Problem 9.2(i) for the proof.]
This theorem may be applied in the following way. The parabolic
proﬁle U
P
(y) of (4.41) has no inﬂexion-point [Fig. 4.3(b)]. Hence it
is not unstable, i.e. neutrally stable. The perturbation solution will
b
U(y)
y
0
Fig. 9.2. (a) No inﬂexion point, (b) inﬂexion-point at y
0
.
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212 Instability and chaos
be characterized by c
i
= 0 and c
r
= 0 within the inviscid theory
according to Prob. 9.2(ii). Namely the perturbation is oscillatory
with a constant amplitude. On the other hand, the proﬁle of a jet has
two inﬂexion-points (Fig. 4.11). This is regarded as unstable within
the inviscid theory,
3
and is actually unstable in real viscous ﬂuids if
the Reynolds number is greater than a critical value R
c
(R
c
≈ 4.0 for
Bickley jet U(y) = sech
2
y see below). As a general feature of ﬂows
of an ideal ﬂuid, if there is a perturbation solution of c
i
< 0 (stable
mode), there exists always another solution of c
i
> 0 (unstable mode)
[Problem 9.2(ii)].
9.3.2. Viscous ﬂows
For viscous ﬂows, the viscosity terms ν∇
2
u
1
and ν∇
2
v
1
must be
added on the right-hand sides of (9.18) and (9.19), respectively.
Hence, the linear perturbation equation (9.20) must be replaced by
(∂
t
+U∂
x
)∇
2
ψ −U

ψ = −∇
2
ψ = ω
1
(9.28)
is the z-component of vorticity perturbation. All the lengths and
velocities are normalized by b and U
m
= max U(y), respectively, with
Re = U
m
b/ν the Reynolds number. It is seen that the above
linear perturbation equation (9.27) is the equation for vorticity
perturbation.
Assuming the same form of normal mode (9.22) for ψ, the above
equation (9.27) becomes
(U −c) (D
2
−k
2
) φ −U

φ −
1
ik Re
(D
2
−k
2
)
2
φ = 0 (9.29)
(D = d/dy). This is known as the Orr–Sommerfeld equation. See
Prob. 9.4(i) for extension to 3D problem.
3
Although the stability analysis here is formulated for channel ﬂows of a ﬁnite
y-width, it is not diﬃcult to extend it to ﬂows in unbounded y axis [DR81].
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9.4. Thermal convection 213
The boundary condition at the walls is the no-slip condition:
φ = 0 and Dφ = 0, at y = −1, 1. (9.30)
Most stability analyses of parallel shear ﬂows of viscous ﬂuids are
studied on the basis of Eq. (9.29).
In general with regard to viscous ﬂows, there exists a certain crit-
ical value R
c
for the Reynolds number Re such that steady ﬂows v
0
are stable when Re < R
c
and becomes unstable when Re exceeds
R
c
. This critical value R
c
is called the critical Reynolds number
which was calculated for various ﬂows:
R
c
≈ 5772 for the 2D Poiseuille ﬂow u
P
(y) of (4.41);
R
c
≈ 520 for the Blasius ﬂow u = f

(y) of Problem 4.4;
R
c
≈ 4.02 for the Bickley jet u(y) = sech
2
y of Problem 4.5.
Furthermore, it is known that R
c
= ∞ (i.e. stable) for the plane
Couette ﬂow u
C
(y) of (4.40), and R
c
= 0 (i.e. unstable) for the
plane shear ﬂow u(y) = tanh y. It is remarked, in particular, that the
Hagen–Poiseuille ﬂow (i.e. the axisymmetric Poiseuille ﬂow) is stable
(i.e. R
c
= ∞), namely that the imaginary part c
i
of the eigenvalue
is negative for all disturbance modes [DR81; SCG80].
The next stability problem is the thermal convection, which
enables us to conduct detailed analysis of chaotic dynamics.
9.4. Thermal convection
9.4.1. Description of the problem
When a horizontal ﬂuid layer is heated from below, ﬂuid motion is
driven by the buoyancy of heated ﬂuid. However, there exist some
opposing eﬀects to control such a motion. They are viscosity and
thermal conductivity. Therefore, it may be said that thermal con-
vections are interplays between the three eﬀects: buoyancy, viscosity
and thermal conduction.
A problem which is historically well studied is the stability anal-
ysis of a horizontal ﬂuid layer heated from below and cooled from
above. Suppose that the temperature diﬀerence between the lower
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214 Instability and chaos
and upper walls is denoted by ∆T. As long as ∆T is small, the
ﬂuid layer remains static and is stable. However, once ∆T becomes
larger than a certain value, the static layer becomes unstable and
convective motion is set up. This kind of ﬂuid motion is called ther-
mal convection. Since the driving force is the buoyancy force, the
temperature dependence of ﬂuid density is one of the controlling
factors.
In order to simplify the problem, we consider a thin horizontal
ﬂuid layer, and the ﬂow ﬁeld is assumed to be two-dimensional with
the x axis taken in the horizontal direction and the y axis taken ver-
tically upward. We apply the Boussinesq approximation (considered
in Sec. 8.4 partly) for the equations of motion. The velocity v is
represented as (u, v). One aspect of the Boussinesq approximation is
that the velocity ﬁeld is assumed to be divergence-free, div v = 0. In
the present case, this reduces to
∂
x
u +∂
y
v = 0. (9.31)
Such two-dimensional divergence-free motions are described by
the stream-function ψ (see Appendix B.2), deﬁning the velocity
components as
u = ∂
y
ψ, v = −∂
x
ψ. (9.32)
Equation (9.31) is satisﬁed automatically by the above.
As the second aspect of Boussinesq approximation, the density ρ
on the left of the equation of motion (8.50) is replaced by a represen-
tative density ρ
0
(constant), and only the density ρ in the buoyancy
term ρg is variable. The acceleration of gravity is given the form
g = (0, −g). Thus, the x-component and y-component (vertical) are
ρ
0
(∂
t
u +u∂
x
u +v∂
y
u) = −∂
x
p +µ∇
2
u, (9.33)
ρ
0
(∂
t
v +u∂
x
v +v∂
y
v) = −∂
y
p +µ∇
2
v −ρg. (9.34)
The density ρ in the buoyancy term is represented as ρ(T), a function
of temperature T only (its pressure-dependence being neglected).
This is because the pressure variation is relatively small since the
ﬂuid layer is assumed very thin. Furthermore, the density is assumed
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9.4. Thermal convection 215
to be a linear function of T as
ρ(T) = ρ
0
(1 −α(T −T
0
)), (9.35)
since the temperature variation is considered relatively small as well,
where α is the thermal expansion coeﬃcient of the ﬂuid, and ρ
0
=
ρ(T
0
) for a representative temperature T
0
.
The equation governing temperature is derived from the general
equation of heat transfer (4.16) of Chapter 4. Suppose that a ﬂuid
particle has gained some amount of heat, and that, as a result, its
temperature is increased by δT together with the entropy increase by
δs. Then the heat gained by the particle is given by a thermodynamic
relation, Tδs = c
p
δT, where the speciﬁc heat c
p
per unit mass at a
constant pressure is used because the heat exchange proceeds in ﬂuid
usually at a constant pressure. Then the entropy term T(Ds/Dt) in
Eq. (4.16) is replaced by the term c
p
(DT/Dt), and we obtain an
equation for the temperature,
ρc
p
(∂
t
T + (v · ∇)T) = σ
(v)
ik
∂
k
v
i
+ div(k grad T). (9.36)
In the stability analysis, the ﬂuid motion is usually very slow. The
ﬁrst term on the right represents the viscous dissipation of kinetic
energy into heat, but it is of second order with respect to the small
velocity and can be ignored because it is of higher order of smallness.
The second term denotes local heat accumulation by nonuniform
heat ﬂux of thermal conduction. This term is taken as the main heat
source, in which the coeﬃcient k is assumed a constant. Thus, we
obtain the equation of thermal conduction as follows:
∂
t
T +u∂
x
T +v∂
y
T = λ∇
2
T, (9.37)
where λ = k/(ρ
0
c
p
) is the coeﬃcient of the thermal diﬀusion.
9.4.2. Linear stability analysis
Suppose that a horizontal ﬂuid layer of thickness d is at rest, and that
the temperature of the lower boundary surface at y = 0 is maintained
at T
1
and that of the upper boundary surface at y = d is T
2
. Hence
the temperature diﬀerence is ∆T = T
1
−T
2
> 0. In the steady state
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216 Instability and chaos
without ﬂuid motion, we can set ∂
t
T = 0, and u = 0, v = 0 in
Eq. (9.37), and obtain
0 = λ∇
2
T = λ

, (9.48)
where (9.46) is used. The right-hand side denotes the rate of gener-
ation of the horizontal z component vorticity ω by the temperature
gradient ∂
x
T

in the horizontal x-direction, called a baroclinic eﬀect
(see Problem 8.2).
The velocity (u, v) satisfying (9.43) is written as u = ∂
y
ψ and
v = −∂
x
ψ, and the vorticity is given by ω = ∂
x
v − ∂
y
u = −∇
2
ψ.
From now, we will use the symbol θ instead of T

for temperature
deformation from the steady distribution T
∗
(y). Then, Eqs. (9.47)
and (9.48) are rewritten as
∂
t
∇
2
ψ −ν∇
2
∇
2
ψ = −gα∂
x
θ, (9.49)
∂
t
θ −λ∇
2
θ = −β∂
x
ψ, (9.50)
where (9.40) is used in (9.50). Thus we have obtained a pair of equa-
tions for ψ and θ, governing the thermal convection.
Now, we normalize the above set of equations by introducing the
following dimensionless variables with a prime,
(x

, y

) = (x/d, y/d), t

= tλ/d
2
,
ψ

= ψ/λ, θ

= θ/(βd).
Substituting these in (9.49) and (9.50) and rearranging the equations,
and dropping the primes of dimensionless variables, we obtain the
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218 Instability and chaos
following set of dimensionless equations:
(∂
t
−∆)θ = −∂
x
ψ, (9.51)
(σ
−1
∂
t
−∆)∆ψ = −R
a
∂
x
θ. (9.52)
where ∆ = ∇
2
(Laplacian), and R
a
is the Rayleigh number and σ
the Prandtl number deﬁned by
R
a
=
gαβd
4
νλ
, σ =
ν
λ
. (9.53)
Furthermore, eliminating θ between (9.51) and (9.52), we ﬁnally
obtain an equation for ψ only:
(∂
t
−∆)(σ
−1
∂
t
−∆)∆ψ = R
a
∂
2
x
ψ. (9.54)
This sixth-order partial diﬀerential equation (∆ is of the second
order) has a simple solution satisfying the following condition of free-
boundary:
ψ = 0, ∂
2
y
ψ = 0, θ = 0 : at y = 0 and y = 1. (9.55)
The boundary conditions (ψ = 0, ∂
2
y
ψ = 0) is called free because the
boundaries (y = 0, 1) are free from the viscous stress, i.e. the stress
vanishes at the boundaries.
4
In addition, the boundaries coincide
with a stream-line by the condition ψ = 0.
The boundary value problem (9.54) and (9.55) has a solution in
the following form:
ψ = Ae
γt
sin(πax) sin(πy), (9.56)
θ = Be
γt
cos(πax) sin(πy), (9.57)
where θ is given to be consistent with (9.51) and (9.56), and
A, B, a, γ are constants. The parameter γ determines the growth
rate of the disturbance ψ (or θ), therefore the stability of the basic
static state.
4
The boundary condition θ = 0 at y = 0, 1 is equivalent to ∂
4
y
ψ = 0 in addition
to ∂
2
y
ψ = 0 by (9.52).
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9.4. Thermal convection 219
In fact, it is not diﬃcult to see that this function ψ satisﬁes the
boundary conditions (9.55). Substituting (9.56) in (9.54), we obtain
the following algebraic equation:
(γ +b
2
)(σ
−1
γ +b
2
)b
2
= R
a
(πa)
2
, (9.58)
where b
2
= π
2
(a
2
+ 1), since we have ∂
2
x
ψ = −(πa)
2
ψ and ∆ψ =
−b
2
ψ. This is a quadratic equation for γ, and it is immediately shown
that γ has two real roots. If γ is negative, the state is stable. If γ is
positive, the state is unstable. The case γ = 0 corresponds to the
neutrally stable state.
The condition of neutral stability is obtained by setting γ = 0
(because γ must be real) in (9.58) as
R
a
(a, γ = 0) = π
4
(a
2
+ 1)
3
/a
2
. (9.59)
Thus, the value of R
a
(a, γ = 0) depends on the horizontal wave
number πa, and has its minimum value given by
R
c
(free) := π
4
(27/4) ≈ 657.5, (9.60)
which is attained at a = a
c
:= 1/
√
2. If the Rayleigh number R
a
is
less than R
c
, we have γ < 0 for all real a, namely the state is stable.
However, if R
a
> R
c
, there exists some range of a in which γ becomes
positive, i.e. the corresponding mode of horizontal wavenumber πa
grows exponentially with time, according to the framework of linear
theory. The number R
c
is called the critical Rayleigh number.
For the case of no-slip boundary condition on solid walls instead
of the free boundary conditions (9.55), the critical Rayleigh number
is known to be
R
c
(no-slip) ≈ 1708,
which is obtained by a numerical analysis (see, e.g. [Cha61]).
9.4.3. Convection cell
It is found from above that for R
a
> R
c
there are some modes (i.e.
a certain range of the wavenumber πa with positive γ) growing
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220 Instability and chaos
exponentially in time,
5
according to the linear stability analysis. As
the unstable modes grow with time, a nonlinear mechanism which
was ineﬀective during small amplitudes now becomes eﬀective and
works to suppress further growth of the modes, and a single mode
of ﬁnite amplitude is selected nonlinearly. Finally, a new equilibrium
steady state will be attained.
6
In the case of the present thermal convection, the nonlinear mech-
anism is favorable to attain a steady equilibrium state. If the relative
diﬀerence (R
a
− R
c
)/R
c
is suﬃciently small, the stream function of
the equilibrium state is given by
ψ
0
(x, y) = A
0
sin(πa
c
x) sin(πy), (9.61)
which corresponds to the stream function (9.56) of the critical
wavenumber a
c
= 1/
√
2 with its amplitude Ae
γt
replaced by a con-
stant A
0
which will be considered in the next section again.
The ﬂow ﬁeld described by the stream function (9.61) is composed
of periodic cells of ﬂuid convection, called thermal convection cells
(Fig. 9.3). Sometimes, the convection cell is called the B´enard cell.
Thermal convection was ﬁrst studied by B´enard (1900) for a liquid
state of heated wax. Later, Rayleigh (1916) solved the mathematical
problem described in the previous section [Sec. 9.4.2]. However, the
y
x
y
g
d
O
T
2
T
1
temperature
(a) Static heat conduction (R
a
<R
c
) (b) Convection (R
a
>R
c
)
Fig. 9.3. Thermal convection cells.
5
From (9.59), R
a
(a, γ = 0) = R
c
+ 9π
4
ξ
2
+ O(ξ
3
), where ξ = a
2
−
1
2
. There is
a range of a
2
of positive γ since we have dγ/dR
a
|
γ=0
= (πa/b
2
)
2
σ/(1 + σ) > 0,
from (9.58).
6
In some instability problems, a nonlinear mechanism is unfavorable for attaining
an equilibrium and leads to explosive growth of modes.
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9.5. Lorenz system 221
convection observed by B´enard is now considered to be the convection
driven by temperature variation of the surface tension of wax, while
the convection solved by Rayleigh is driven by buoyancy. The latter
is often called the Rayleigh convection, while the former is termed
the Marangoni convection.
9.5. Lorenz system
It has been found so far that thermal convection is controlled by
Rayleigh number R
a
and Prandtl number σ within a framework of
linear theory. In this section, we consider a ﬁnite-amplitude con-
vection, which is selected by a nonlinear mechanism among unstable
modes. As the value of the control parameter R
a
is increased further,
the state of steady convection becomes unstable and a new oscilla-
tory mode sets in. This is termed the Hopf bifurcation (Sec. 9.1).
When the Rayleigh number becomes suﬃciently high, and hence the
nonlinearity is suﬃciently large with an appropriate value of Prandtl
number, the time evolution of state can exhibit a chaotic behavior.
One of the historically earliest examples of chaos realized on com-
puter is the Lorenz dynamical system.
9.5.1. Derivation of the Lorenz system
In order to take into account the nonlinear eﬀect appropriately, it is
useful to simplify the expression of convection cells as much as pos-
sible, so that a simplest system of ordinary diﬀerential equations can
be derived. The Lorenz system is such a kind of dynamical system.
Suppose that the stream function ψ and the temperture deformation
θ of convection cells are represented by
ψ(x, y, t) = −AX(t) sin
πa
d
xsin
π
d
y, (9.62)
θ(x, y, t) = BY (t) cos
πa
d
xsin
π
d
y −
B
√
2
Z(t) sin
2π
d
y, (9.63)
A =
√
2λ
1 +a
2
a
, B =
√
2 ∆T
π
R
c
R
a
,
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222 Instability and chaos
where x, y are dimensional variables, and R
c
is given by (9.60): i.e.
R
c
= π
4
(a
2
+ 1)
3
/a
2
with a = 1/
√
2. The time factors e
γt
of (9.56)
and (9.57) are replaced by X(t) and Y (t), respectively, and a new x-
independent term with a time factor Z(t) is added to the temperature
deformation θ. The temperature is given by T = T
∗
(y) + θ, where
T
∗
(y) is the steady distribution (9.39).
The two functions ψ and θ are required to satisfy the full nonlinear
equations of motion. A linear equation for the vorticity ω = ∂
x
v −
∂
y
u = −∇
2
ψ was given by (9.48). Nonlinear version of the vorticity
equation for −∇
2
ψ can be derived by taking curl of the original
governing equations (9.33) and (9.34), and a nonlinear version of
the temperature equation for θ can be derived from (9.37). Both are
written as follows:
∂
t
∇
2
ψ + (∂
y
ψ∂
x
−∂
x
ψ∂
y
)∇
2
ψ = ν∇
2
∇
2
ψ −gα∂
x
θ, (9.64)
∂
t
θ + (∂
y
ψ∂
x
−∂
x
ψ∂
y
)θ = λ∇
2
θ −β∂
x
ψ. (9.65)
The Lorenz system is obtained by substituting the expressions (9.62)
and (9.63). In the resulting equations, only the terms of the forms,
sin
πa
d
x sin
π
d
y, cos
πa
d
x sin
π
d
y, sin
2π
d
y,
are retained in the equation, and the other terms are omitted. Equat-
ing the coeﬃcients of each of the above terms on both sides, we obtain
the following dynamical equations of Lorenz system (Problem 9.3):
dX/dτ = −σX +σY,
dY/dτ = −XZ +rX −Y,
dZ/dτ = XY −bZ,
(9.66)
where
r =
R
a
R
c
, σ =
ν
λ
, b =
4
1 +a
2
, (9.67)
and τ = λt(1 + a
2
)π
2
/d
2
is the normalized time variable, which is
again written as t (in order to follow the tradition). The right-hand
sides of the above system of equations do not include time t. Namely,
the time derivatives on the left-hand sides are determined solely by
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9.5. Lorenz system 223
the state (X, Y, Z). Such a system is called an autonomous system,
in general.
The trajectory (X(t), Y (t), Z(t)) of state is determined by inte-
grating the above system (9.66) numerically. A set of points in the
phase space (X, Y, Z) where a family of trajectories for a set of ini-
tial conditions accumulate asymptotically as t → ∞ is called an
attractor.
We can think of time derivatives
˙
X = dX/dt,
˙
Y = dY/dt,
˙
Z = dZ/dt as three components of a velocity vector V deﬁned at
the point X = (X, Y, Z). Then we can regard the system dynam-
ics like a ﬂuid motion with velocity ﬁeld V(X). An important fea-
ture of such a motion of points in the phase space (X, Y, Z) is the
property that the phase volume composed of points moving with
V = (
˙
X,
˙
Y ,
˙
Z) decreases steadily. This is veriﬁed simply as fol-
lows. Taking the divergence of velocity ﬁeld (
˙
X,
˙
Y ,
˙
Z), we obtain
div V = ∂
˙
X/∂X + ∂
˙
Y /∂Y +∂
˙
Z/∂Z = −σ −1 −b < 0. Due to this
property, the phase volume of an attractor where the trajectories are
approaching diminishes indeﬁnitely. This does not necessarily mean
that the attractors are points, but means only that the dimension of
the attractor is less than 3. This will be remarked again later. Here it
is mentioned only that such a shrinking of phase volume is a common
nature of dissipative dynamical systems, in which the kinetic energy
is transformed to heat (say).
9.5.2. Discovery stories of deterministic chaos
There is an interesting story about the discovery of this system by
Lorenz himself. He tried to solve an initial value problem of the
above system (9.66) of ordinary diﬀerential equations numerically
by using a calculator which was available for him at the time of
1960s, where the initial values were (X(0), Y (0), Z(0)) = (0, 1, 0)
with σ = 10, r = 28, b = 8/3. Needless to say, it took much time for
calculations.
He wanted to obtain results at times further ahead. Without using
the above initial values, he used the values obtained at an interme-
diate time of computation, and carried out the calculation with the
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224 Instability and chaos
Fig. 9.4. Curve X(t) obtained numerically by integrating the Lorenz system
(see Sec. 9.5.2 for the details). Curves A and B are obtained from two slightly
diﬀerent initial conditions.
same computer program. It was expected that the computer would
generate results of approximately same data for overlapping times.
At this computation, he thoughtlessly used only the ﬁrst three dig-
its instead of the six digits data obtained by the calculation (e.g.
curve A of Fig. 9.4). Naturally, at the beginning, the computer gave
similar temporal behavior as before. However, as time went on, both
results began to show diﬀerences, and ﬁnally showed a totally dif-
ferent time evolution from the previous computation (e.g. curve B
of Fig. 9.4). Reﬂecting on those times in his study, he wrote that he
had been shocked at this outcome. He was already convinced that
the set of above equations captured a certain intrinsic chaotic nature
of atmospheric phenomena, and that long-term weather prediction is
doubtful.
Chaotic dynamical systems are characterized to have a property
that they are sensitively dependent on initial conditions. The above
result observed by Lorenz is just proved this fact. After detailed
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9.5. Lorenz system 225
analysis made at later times, the temporal behaviors of the Lorenz
system are found to be stochastic in a true sense, and thus long-term
behaviors of its solution are unpredictable [Lor63].
Despite the fact that the time evolution of a system state is
determined by a set of equations, so that its short-term evolution is
deterministic, it is found that its long-term behaviors are stochastic.
This phenomenon is called the deterministic chaos. This was found
by detailed analysis of the chaos-attractor in [Lor63], now called
the Lorenz attactor. Many nonlinear dynamical systems have such
property.
In 1961, earlier than the Lorenz study, another chaos-attractor
had been discovered by Y. Ueda [Ued61] during his study of analog-
computer simulations of nonautonomous nonlinear system (corre-
sponding to the Van der Pol equation under a time-periodic external
forcing with a third-order term). Reminding of the long hours sit-
ting in front of the analog-computer (made by his senior colleague
M. Abe), Ueda writes “After those long exhausting vigils in front of
the computer, staring at its output, chaos had become a totally natu-
ral, everyday phenomenon in my mind.” This is called Ueda attactor
[Ued61].
Solutions of the Navier–Stokes equation are believed also to have
such behavior in turbulent states which we will consider in the next
chapter.
9.5.3. Stability of ﬁxed points
In stability analysis, steady states play an important role, because
the growth or decay of perturbations to a steady state is usually
investigated. A steady state corresponds to a ﬁxed point in the
dynamical system. A ﬁxed point of the Lorenz system (9.66) is deﬁned
by the point where the velocity V vanishes, i.e. (
˙
X,
˙
Y ,
˙
Z) = 0. Set-
ting the right-hand sides of (9.66) equal to 0, we obtain
X = Y, −XZ +rX −Y = 0, XY −bZ = 0.
For r = R
a
/R
c
< 1, we have only one ﬁxed point which is the
origin O: (X, Y, Z) = (0, 0, 0). The point O corresponds to the static
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226 Instability and chaos
state. This is reasonable because we are considering a bifurcation
problem from the static state to a convection-cell state. For r =
R
a
/R
c
> 1, it can be shown without diﬃculty that there exist three
ﬁxed points: (a) the origin O: (0, 0, 0), and (b) two points C and C

deﬁned by
C : (q, q, r −1), C

: (−q, −q, r −1), when r > 1,
where q =

b(r −1). The ﬁxed points C and C

are located at points
of mirror symmetry with respect to the vertical plane X + Y = 0.
Now, let us consider the stability of the three ﬁxed points.
When r > r
c
(r
c
> 1, a constant), we will encounter a strange
situation in which all the three ﬁxed points are unstable, in other
words we have no stable ﬁxed points to which the trajectory should
aproach. This is a situation where the moving state has no point to
head towards and trajectories become chaotic.
9.5.3.1. Stability of the point O
Linearizing the Lorenz system (9.66) for points in the neighborhood
of O, we obtain
˙
X

= −σX

+σY

,
˙
Y

= rX

−Y

,
˙
Z

= −bZ

,
where X

, Y

, Z

are small perturbations to the ﬁxed point (0, 0, 0),
which are represented as X

and substituting
those in the above equations, we obtain a system of linear algebraic
equations, i.e. a matrix equation, for the amplitude (X
0
, Y
0
, Z
0
):

¸
¸
s +σ −σ 0
−r s + 1 0
0 0 s +b
¸

¸
¸
X
0
Y
0
Z
0
¸

= 0.
In order that there exists a nontrivial solution of (X
0
, Y
0
, Z
0
), the
determinant of the coeﬃcient matrix must vanish. Thus, we obtain
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9.5. Lorenz system 227
an eigenvalue equation for the growth rate s, which is also termed
the characteristic exponent:

s +σ −σ 0
−r s + 1 0
0 0 s +b

= (s +b)

s
2
+ (σ + 1)s +σ(1 −r)

= 0.
Therefore, we have
s
0
= −b, s
±
= −
1
2
(σ + 1) ±
1
2

(σ + 1)
2
+ 4σ(r −1).
If r < 1, the point O is stable, i.e. all Re(s
0
) and Re(s
±
) are neg-
ative. This is consistent with the linear theory because the point O
corresponds to the static state which is stable for r = R
a
/R
c
< 1.
However, once r > 1, we have Re(s
+
) > 0. Thus, it is found that the
point O becomes unstable for r > 1, which is also consistent with
the linear theory.
9.5.3.2. Stability of the ﬁxed points C and C

is that
r > 1. As long as r − 1 = ε is suﬃciently small (and positive), it
is found that all three roots are negative. Hence the two points C
and C

are stable as long as ε is small. This corresponds to a stable
steady convection. The two ﬁxed points denote two possible senses
of circulatory motions within a cell.
Setting r = 1 (when both C and C

happen to coincide with O), it
can be shown immediately that the three roots are 0, −b, −(σ + 1),
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228 Instability and chaos
which are assumed to correspond to s
1
, s
2
, s
3
, respectively in order.
As r increases from 1, it can be shown from a detailed analysis
(not shown here) that s
1
decreses from 0 and tends to be com-
bined with s
2
, i.e. s
1
= s
2
< 0 at a certain value of r. Thereafter,
both together form a complex-conjugate pair, and their real part
increases and reaches zero in due course. From there, the points
C and C

become unstable. Namely, an instability sets in when
Re(s
1
) = Re(s
2
) = 0. At that critical stability point, we must have
s
1
= +iω, and s
2
= −iω (where ω is real). As far as r > 1, s
3
is always
negative. At the critical point, we have s
1
+ s
2
= 0. On the other
hand, from the cubic equation F(s) = 0, we obtain the sum of three
roots given by s
1
+s
2
+s
3
= s
3
(at the critical point) = −(σ+b +1).
Since s
3
is one of the roots, we have the following equation,
0 = F(−(σ +b + 1)) = rb(σ −b −1) −bσ(σ +b + 3),
at the critical point. Therefore, denoting the value of r satisfying this
equation by r
c
, we obtain
r
c
:=
σ(σ +b + 3)
σ −b −1
.
We have already seen that the ﬁxed point O is unstable for r > 1.
Now it is veriﬁed that other ﬁxed points C and C

become unstable
as well for r > r
c
. When r
c
> 1 for given values of σ and b, we
obtain that all three ﬁxed points are unstable for r > r
c
, in other
words, we have no stable ﬁxed points to which the trajectory should
approach. Now, we have a situation where the state point has no
ﬁxed direction. This is understood as a favorable circumstance for
chaotic trajectories.
Another useful property can be veriﬁed by deﬁning a non-negative
function, H(X, Y, Z) := X
2
+Y
2
+(Z −r −σ)
2
. It can be shown by
using the Lorenz system (9.66) that, if (X
2
+Y
2
+Z
2
)
1/2
is suﬃciently
large,
d
dt
H =
d
dt
(X
2
+Y
2
+ (Z −r −σ)
2
) < 0.
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9.6. Lorenz attractor and deterministic chaos 229
Hence, the trajectory in the phase space at large distances from the
origin is moving inward so as to reduce the distance between (X, Y, Z)
and (0, 0, r + σ). This sort of a positive-deﬁnite function such as
H(X, Y, Z) is called a Liapounov function and plays an important
role in the global stability analysis as given above.
9.6. Lorenz attractor and deterministic chaos
9.6.1. Lorenz attractor
From the analysis of the Lorenz system of diﬀerential equations in
the previous section, we have found the following properties.
(i) There is no stable ﬁxed points for r > r
c
.
(ii) Trajectories are directed inward at large distances from the
origin.
(iii) Phase volume shrinks steadily during the orbital motion.
(iv) There is no repeller (see the footnote), no unstable limit-cycle,
and no quasi-periodic orbit.
The property (iv) results from the negative value of the sum s
1
+
s
2
+s
3
. The sum s
1
+s
2
+s
3
must be positive for both the repeller
and unstable limit-cycle.
7
If there existed a quasi-periodic orbit, it
contradicts with the property (iii) because the phase volume should
be conserved in the quasi-periodic motion.
Then, where should the trajectories head for? Anyway a brief time
direction can be determined by the system of diﬀerential equations
(9.66). The trajectory of Fig. 9.5 was computed numerically for the
case r > r
c
with parameter values,
r = 28, σ = 10, b = 8/3,
7
The repeller is a ﬁxed point whose characteristic exponents are all positive,
while the unstable limit-cycle is a periodic orbit having two positive characteristic
exponents and one zero exponent corresponding to the direction of the periodic
orbit.
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230 Instability and chaos
Fig. 9.5. Curve Y (t) obtained numerically by integrating the Lorenz system.
for which r
c
= 24.74, and C and C

are (±8.48, ±8.48, 27). The
object to which the trajectories draw asymptotically is now called
the Lorenz attractor (Fig. 9.6).
Trajectories of a dynamical system characterized by sensitive
dependence on initial conditions, i.e. chaotic trajectories, accumu-
late in the limit as t → ∞ at a manifold of a fractal dimension, in
general. Such a set of points is called a strange attractor. The concept
of strange is related not only to such a geometry, but also to an orbital
dynamics. Lyapunov characteristic exponents are always deﬁned for
a dynamical system with their number equal to the dimension of the
dynamical system. If the trajectory has at least one positive expo-
nent (corresponding to one positive s), then the manifold composed
of these trajectories is called a strange attractor.
The Lorenz attractor is a representative example of the strange
attractor. Along the trajectory, the phase volume extends in the char-
acteristic direction of positive exponents, whereas it reduces in the
direction of negative exponents, and as a whole, the phase volume
shrinks along the trajectory to a phase volume of dimension less
than 3 in the case of the Lorenz system. As a result, the strange
attractor becomes fractal.
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9.6. Lorenz attractor and deterministic chaos 231
Fig. 9.6. Lorenz attractor: projection of a trajectory to (a) (X, Z)-plane,
(b) (X, Y )-plane, and (c) (Y, Z)-plane (see Sec. 9.6.1).
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232 Instability and chaos
9.6.2. Lorenz map and deterministic chaos
In order to observe the time evolution of orbits, we consider the Z-
coordinate of the orbit projected in the (Y, Z)-plane, which repeats
up-and-down motions. Denoting the nth upper-extremum of Z-
coordinate by M
n
and the next upper-extremum by M
n+1
, we create
a pair (M
n
, M
n+1
) for a number of n’s.
Taking M
n
as the coordinate along the horizontal axis and M
n+1
as the coordinate along the vertical axis, one can plot a point
(M
n
, M
n+1
) on the (M
n
, M
n+1
) plane. Plotting those points for a
number of n’s, one can observe those points distribute randomly
along a curve of Λ-form (Fig. 9.7), not spread over an area in
the (M
n
, M
n+1
)-plane. This remarkable property was discovered by
Lorenz and this plot is called the Lorenz map. Apparently the
irregular behavior of the orbit exhibits a certain regularity (lying on
a one-dimensional object). However, this property itself is evidence
Fig. 9.7. Lorenz map.
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9.6. Lorenz attractor and deterministic chaos 233
that the long-term evolution of the orbit is unpredictable, and its
behavior is chaotic. Let us look at this property with a simpliﬁed
but similar tent-map without losing the essential property of the
Lorenz map.
The tent-map (looking like a tent, Fig. 9.8) is deﬁned by
x
n+1
= Tx
n
=

2x
n

0 < x
n
<
1
2

2(1 −x
n
)

1
2
< x
n
< 1

where x
n
corresponds to (M
n
− m)/(M − m) with m and M being
the minimum and maximum of M
n
, respectively.
Let us express x
n
by binary digits. If x
1
= 0.a
1
a
2
a
3
a
4
· · · by the
binary, that denotes the following:
x
1
= a
1
2
−1
+a
2
2
−2
+a
3
2
−3
+a
4
2
−4
+· · · ,
where a
i
takes 0 or 1. In addition, we can represent 1 − x
1
as
follows:
1 −x
1
= 0.1111 · · · −0.a
1
a
2
a
3
a
4
· · ·
= 0.(Na
1
)(Na
2
)(Na
3
)(Na
4
) · · ·
Fig. 9.8. Tent map.
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234 Instability and chaos
where Na = 1 −a. According to this, a T-operation becomes
Tx
1
= 0.(N
a
1
a
2
)(N
a
1
a
3
)(N
a
1
a
4
) · · ·
where N
a
b = b (when a = 0), Nb (when a = 1). Namely, after an
operation of the map T, each digit moves to the left up-shift by one
place. The ﬁrst digit moves up to the left of the “decimal” point and
becomes 0, indicating that the number is less than 1. This is carried
out as follows. If a
1
= 0, the point x is located on the left-half of the
horizontal axis of the tent-map, and we have N
a
1
a
1
= a
1
= 0 and
N
a
1
a
k
= a
k
(for k = 2, . . .). If a
2
= 0 (a
1
= 0), then the map Tx
1
is
located on the left-half of the horizontal axis.
If a
1
= 1, the point x is located on the right-half of the horizontal
axis, and we have N
a
1
a
1
= Na
1
= 0 and N
a
1
a
k
= Na
k
(for k = 2, . . .).
If a
2
= 0 (a
1
= 1), then N
a
1
a
2
= Na
2
= 1, and the map Tx
1
is located on the right-half of the horizontal axis. Repeating this
operation n times, we obtain
T
n
x
1
= 0.(Pa
n+1
)(Pa
n+2
)(Pa
n+3
) · · ·
where P = N
a
1
+a
2
+··· +a
n
.
Suppose that we are given two numbers x
1
and x

1
whose ﬁrst
six digits are identical with 0.110 001. Regarding x
1
, the digits are
all 0 at the seventh place and thereafter, while for the number x

1
the
digits take either 0 or 1 randomly after the seventh place, such as
x
1
= 0.110 001 000 · · · , x

1
maps are considered to coincide practically.
After T
6
map, these numbers behave in a totally diﬀerent way. In par-
ticular, the behavior of T
n
x

1
(n ≥ 6) will be unpredictable, because
the digits are a random sequence by the presupposition.
The property that cut-oﬀ errors move up-shift to the left for each
map corresponds to the exponential growth of error, i.e. exponential
instability. Therefore in the dynamical system of the tent-map, the
orbit of T
n
is sensitive with respect to the accuracy of initial data,
and becomes unpreditable after a certain n, say n
0
(n
0
= 6 in the
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9.7. Problems 235
above example). Such a kind of motion is called a deterministic
chaos.
In the numerical experiment of the Lorenz system with r = 28,
σ = 10, b = 8/3, two orbits starting from two neighboring points
near the Lorenz attractor will lose their correlation before long. Such
a property of sensitive dependence on initial values is a characteristic
property of chaotic orbits. After the time, either of the orbits traces
a path on the same attractor without mutual correlation.
The Lyapunov dimension (a kind of fractal dimension) of the
Lorenz attractor for the above parameter values is estimated as
2.0 < D
L
< 2.401.
Because of this fractal dimension, the Lorenz attractor is a strange
attractor.
9.7. Problems
Problem 9.1 Rayleigh–Taylor instability
Suppose that a heavy ﬂuid of constant density ρ
1
is placed above a
light ﬂuid of another constant density ρ
2
, and separated by a surface
S in a vertically downward gravitational ﬁeld of acceleration g. In
the unperturbed state, the surface S was a horizontal plane located
at y = 0 (with the y axis taken vertically upward) and the ﬂuid was
at rest, and the density was
ρ
1
for y > 0; ρ
2
(< ρ
1
) for y < 0.
Suppose that the surface S is deformed in the form,
y = ζ(x, t), (9.68)
[Fig. 9.9(a)]. Both above and below S, the ﬂow is assumed to be
irrotational and the velocity potential φ is expressed as
φ = φ
1
(x, y, t) for y > ζ; φ
2
(x, y, t) for y < ζ. (9.69)
(i) Derive linear perturbation equations for small perturbations ζ,
φ
1
and φ
2
from the boundary conditions (6.8) and (6.10).
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236 Instability and chaos
Fig. 9.9. (a) Rayleigh–Taylor instability, (b) internal gravity wave.
(ii) Expressing the perturbations in the following forms with a
growth rate σ and a wavenuumber k,
ζ = Ae
σt
e
ikx
, φ
i
= B
i
e
σt
e
ikx−ky
, (i = 1, 2), (9.70)
derive an equation to determine the growth rate σ, where
A, B
1
, B
2
are constants. State whether the basic state is sta-
ble or unstable.
(iii) Apply the above analysis to the case where a lighter ﬂuid is
placed above a heavy ﬂuid, and derive a conclusion that there
exists an interfacial wave (called the internal gravity wave,
Fig. 9.9(b)). State what is the frequency.
Problem 9.2 Rayleigh’s inﬂexion-point theorem
Based on the Rayleigh’s equation (9.23) or (9.26), verify the
followings:
(i) We have the Rayleigh’s inﬂexion-point theorem stated in
Sec. 9.3.1.
(ii) If there is a stable solution of c
i
< 0 (stable mode), there always
exists an unstable solution too, for inviscid parallel ﬂows.
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9.7. Problems 237
Problem 9.3 Lorenz system
Derive the Lorenz system (9.66) from the system of Eqs. (9.64) and
(9.65) by using the stream function ψ and the temperture deforma-
tion θ deﬁned by (9.62) and (9.63).
Problem 9.4 Squire’s theorem
Suppose that there is a steady parallel shear ﬂow v
0
= (U(z), 0, 0) of
a viscous incompressible ﬂuid of kinematic viscosity ν in a channel
−b < z < b in the cartesian (x, y, z) space.
(i) Derive the following normalized perturbation equations
∇
2
u −iαR
e
(U(z) −c)u −R
e
U

α
2
+β
2
.
(iii) Show that the critical Reynolds number R
2D
c
for two-dimen-
sional disturbances is always less than the critical Reynolds
number R
3D
c
for three-dimensional disturbances [Squire’s
theorem].
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Chapter 10
Turbulence
Turbulence is an irregular ﬂuctuating ﬂow ﬁeld both spatially and
temporally. It is a fundamental problem to be answered whether the
ﬂuid mechanics of continuum media can capture the turbulence.
In the past, there have been a number of discussions about how
such turbulence is generated. On the one hand, turbulence is con-
sidered to be generated by spatially random initial conditions, or
temporally random boundary conditions. This is a view of turbu-
lence generated passively. In the modern view, however, turbulence
is considered to be an irregularly ﬂuctuating ﬂow ﬁeld which devel-
ops autonomously by a nonlinear mechanism of ﬁeld dynamics. This
change in view on turbulence is largely due to the recent development
of the theory of chaos considered in the last chapter (Chapter 9).
Chaos is studied mostly for nonlinear dynamical systems of low
dimensions. In a chaotic state, an initial slight diﬀerence of state
grows exponentially with time. Despite the fact that the system
evolves according to a governing equation, the state of the dynamical
system becomes unpredictable after a certain time. This is a property
termed deterministic chaos considered in Chapter 9.
Turbulence is a dynamical system of considerably many degrees
of freedom. It is not surprising that the turbulence exhibits much
more complex behaviors. In such a (turbulence) ﬁeld, it would not
be realistic to specify the initial condition accurately, and a statisti-
cal consideration must be made. There are some intrinsic diﬃculties
in statistics: that is, statistical distributions of velocity or velocity
diﬀerence at two points are not the Gaussian distribution and hence
239
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240 Turbulence
are not normal. This means we have to take into account all the
statistical moments of variables.
Here, we take a view that turbulence is a physical system of
nonlinear dynamics of a continuous ﬁeld with a dissipative mech-
anism. In other words, turbulence exhibits every possible ﬂuctuating
motions and straining ﬂows without breaking the fundamental laws
of mechanics of a continuous medium. Under such a view, we investi-
gate turbulence dynamics governed by the Navier–Stokes equations
(4.5) and (4.6) for the velocity ﬁeld u
i
of an incompressible ﬂuid of
constant density ρ:
∂
t
u
i
+ ∂
j
(u
i
u
j
) = −ρ
−1
∂
i
p + 2ν∂
j
e
ij
, (10.1)
D = ∂u
i
/∂x
i
= 0, (10.2)
where ν is the kinematic viscosity, p the pressure and e
ij
=
1
2
(∂
i
u
j
+
∂
j
u
i
).
1
When the statistical properties of turbulence are uniform in space,
that is, the statistics are equivalent at every point of space, turbu-
lence is called homogeneous. In addition, if the statistics of turbulence
are equivalent for any direction of space, that is, the statistics such
as correlation functions at n points (n: some integer) are invariant
with respect to rigid-body rotation or inversion of those points, then
the turbulence is called isotropic.
10.1. Reynolds experiment
It is generally accepted that one of the earliest studies of turbulence is
the Reynolds experiment (1883). His experimental tests were carried
out for ﬂows of water through glass tubes of circular cross-sections of
diﬀerent diameters. The ﬂow through a circular tube of diameter D is
characterized with maximum velocity U. In Sec. 4.4, we learned that
the ﬂow is controlled by a dimensionless parameter, i.e. the Reynolds
number deﬁned by Re = UD/ν, where ν is the kinematic viscosity
(of water). This view owes a great deal to the study of Reynolds.
1
Note that ∂
j
(u
i
u
j
) = u
j
∂
j
u
i
and ∂
j
2e
ij
= ν∇
2
u
i
, since ∂
j
u
j
= 0 and ∂
2
j
= ∇
2
.
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10.1. Reynolds experiment 241
According to his experiment, there is a critical value R
c
for the
Reynolds number. Below R
c
, the ﬂow is laminar, i.e. the water ﬂows
smoothly along the whole length of the tube. If the ﬂow velocity u
along the tube is represented as a function u(r) of the radial coor-
dinate r with r = 0 denoting the tube axis, the smooth ﬂow takes a
parabolic proﬁle u(r) = U(1 − r
2
/a
2
), which is called the Poiseuille
ﬂow (see (4.78) for axisymmetry;
2
(4.41) for 2D problem). Above
R
c
, he observed turbulent ﬂows. An important point which Reynolds
observed is the existence of the critical value R
c
of the Reynolds
number for the transition to turbulence. He also observed that the
friction laws are diﬀerent for ﬂows below and above R
c
. The friction
was proportional to U below R
c
, while it was proportional to U
2
above R
c
. This suggests that the two types of ﬂows are diﬀerent in
nature.
As Reynolds number becomes suﬃciently high, not only pipe ﬂows
but most ﬂows become turbulent. At each point of turbulence ﬁeld,
ﬂow velocity ﬂuctuates with time. Taking an average of the irregu-
larly ﬂuctuating velocity vector u = (u
i
) at a point x = (x
i
), we
obtain an average velocity ﬁeld 'u` = 'u
i
`. The average velocity
'u`(x) may vary slowly and smoothly from point to point, while the
diﬀerence u

= u − 'u` may ﬂuctuate irregularly, with its average
'u

` vanishing. u

denotes the turbulent ﬂuctuation of velocity.
Another remarkable contribution of Reynolds in addition to the
experiment mentioned above is the introduction of an average equa-
tion and recognition of “Reynolds stress” in it. Suppose that the
velocity and pressure ﬁelds of an incompressible ﬂuid are represented
by superposition of an average and a ﬂuctuation as follows:
u
i
= 'u
i
`(x, t) + u

j
` (multiplied by ρ) is called the Reynolds stress,
because this can be combined with the viscous stress term on the
right as
ρ∂
j
(2ν'e
ij
` −'u

i
u

j
`) = ∂
j
σ
(turb)
ij
,
where σ
(turb)
ij
≡ 2µ'e
ij
` −ρ'u

i
u

j
`, where µ = ρν. Thus, we obtain the
same form as (10.1) for the equation of average ﬁelds.
10.2. Turbulence signals
In turbulent ﬂows, velocity signals ﬂuctuate irregularly with time
[Fig. 10.1(a)]. In fact, there are a large number of modes from very
small scales to very large scales in turbulence, which interact with
each other. Their spatial distributions are also considered to be ran-
dom. Statistical distribution of each (x, y, z) component of u

may be
expected to be nearly Gaussian. However, remarkably enough, real
turbulences are not so, and exhibit non-Gaussian properties for sta-
tistical distributions [Fig. 10.1(c)]. This implies that there may exist
some nonrandom objects in turbulence ﬁeld to make it deviate from
a complete randomness of Gaussian statistics.
In general, a skewness factor of the distribution of longitudinal
derivative
3
of a turbulent velocity-component u, denoted by u
x
=
∂u/∂x, is deﬁned by
Skewness S[u
x
] =
'(u
x
)
3
`
'(u
x
)
2
`
3/2
(≈ −0.4 in an experiment),
which is always found to be negative. In the Gaussian distribution,
the skewness must vanish. Furthermore, a ﬂatness factor of u
x
is
3
The longitude denotes that the direction of derivative is the same as the velocity
component u, not lateral to it.
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10.2. Turbulence signals 243
Fig. 10.1. (a) Fluctuating signal of x component u of turbulent velocity u

,
(b) Enlargement of the interval A of (a). (c) A sketch of non-Gaussian distribution
function P(u
x
) of the longitudinal derivative u
x
= ∂
x
u, where the broken curve
is the Gaussian distribution function [Mak91].
deﬁned by
Flatness F[u
x
] =
'(u
x
)
4
`
'(u
x
)
2
`
2
(≈ 7.2 in an experiment).
This is also deviating from the Gaussian value 3.0.
Nonrandom objects in the turbulence ﬁeld causing these devia-
tions from Gaussian distribution are often called organized structures.
Recent direct numerical simulations (DNS in short) of turbulence at
high Reynolds numbers present increasing evidence that homoge-
neous isotropic turbulence is composed of random distribution of
a number of long thin vortices, often called worms. Their cross-
sectional diameters are estimated to be of the order of l
d
(deﬁned
by (10.18) below), and their axial lengths are much larger.
4
From some experimental observation of the probability distribu-
tion of u
x
, it is found that the probability of large [u
x
[ is relatively
4
Some computer simulation showed that the average separation distance between
worms is of the order of Taylor micro-scale deﬁned by a length scale of statistical
correlation of two velocities at two diﬀerent points.
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244 Turbulence
higher than that of the Gaussian. Namely, large derivative values
u
x
appears intermittently in turbulence. This is one of the charac-
teristic features of ﬂuid turbulence and is called intermittency. It
is not only observed by experiments, but also found by computer
simulations.
10.3. Energy spectrum and energy dissipation
10.3.1. Energy spectrum
Let us consider a turbulent ﬂow ﬁeld in a three-dimensional space.
It is convenient to consider turbulence in Fourier space. In order
to avoid some mathematical diﬃculties arising from the inﬁnity of
space, the ﬂow ﬁeld is assumed to satisfy a periodic condition. Sup-
pose that the velocity ﬁeld u(x, y, z) is periodic with respect to the
three cartesian directions (x, y, z) of periodicity length L:
u(x + m
x
L, y + m
y
L, z + m
z
L) = u(x, y, z),
where m
x
, m
y
, m
z
∈ Z (integer). The pressure ﬁeld p(x, y, z) is sim-
ilarly represented. Hence, the ﬂow ﬁeld is considered in a basic cubic
space:
( : [0 x < L, 0 y < L, 0 z < L].
The representation in an inﬁnite space can be obtained by taking the
limit, L →∞. Then the velocity ﬁeld u(x, t) is given by the following
representation:
u(x, t) =

R
3
Φ(k, t)d
3
k, Φ(k, t) =
1
2
[ˆ u(k, t)[
2
. (10.12)
If we take the average with respect to a statistical ensemble of ini-
tial conditions in addition to the average over the space (, then the
average ' `
C
should be replaced with the ensemble average ' `.
Provided that the turbulence is regarded as isotropic and the aver-
age 'Φ(k, t)` depends on the magnitude k = [k[ of the wavenumber
k, then we have

u
2
2

(t) =

∞
0
E(k, t)dk, E(k, t) = 4πk
2
Φ([k[, t), (10.13)
where E(k, t) is the energy spectrum at a time t.
One of the advantages of Fourier representation is that we can
consider each Fourier component separately. The component of a
wavenumber k corresponds to the velocity variation of wavelength
2π/k. Such a component is said to have simply a length scale l = 1/k.
It is often called an eddy of scale l. Furthermore, the Fourier represen-
tation with respect to space variables (x, y, z) enables us to reduce a
partial diﬀerential equation to an ordinary diﬀerential equation with
respect to time t, since ∂
x
is replaced by ik
x
, etc.
10.3.2. Energy dissipation
Let us consider an energy equation. With an analogous calcula-
tion carried out in deriving Eq. (4.23) for unbounded space in
Sec. 4.3, an energy equation is written for the kinetic energy
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10.3. Energy spectrum and energy dissipation 247
K =

∞
0
k
2
E(k)dk. [See Problem 10.1] (10.17)
The expressions on the right-hand side give three diﬀerent expres-
sions of energy dissipation. The form of integrand k
2
E(k) of (10.17)
implies that the dissipation rate is ampliﬁed for eddies of large k
(small eddies) due to the factor k
2
.
10.3.3. Inertial range and ﬁve-thirds law
A number of experiments have been carried out so far in order to
determine the energy spectrum E(k) for turbulence at very high
Reynolds numbers, 10
4
or larger. In fully developed turbulence, most
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248 Turbulence
Fig. 10.2. Log-log plot of an experimentally determined energy spectrum E
1
(k),
showing the inertial range by a straight-line slope of −5/3 (from a grid turbulence
with a main stream velocity 5 m/s in the x-direction [Mak91], where E
1
(k) is the
one-dimensional spectrum which is connected to the three-dimensional energy
spectrum E(k) by E(k) = k
3
(d/dk)(k
−1
dE
1
(k)/dk) (if E
1
(k) ∝ k
α
, then E(k) ∝
k
α
too). [See Problem 10.3.] A
L
and A
T
are obtained from a fully developed
turbulence, while B
L
and B
T
are from a nondeveloped turbulence. The suﬃces L
and T denote longitudinal (x) and transverse components (y and z) of velocity.
energy spectra determined experimentally suggest the form,
E(k) ∝ k
−5/3
for an intermediate range of wavenumbers (an inertial range). The
range is deﬁned by wavenumbers which are larger than k
Λ
associ-
ated with a scale Λ (a laboratory scale) of external boundaries and
less than k
d
associated with a scale of viscous dissipation. This is
known as the 5/3-th power law of the energy spectrum. A plot of
log E(k) versus log k shows a nearly straight-line behavior of slope
−5/3 in the corresponding range of k (Fig. 10.2). Such a range of
wavenumbers is called the inertial range. Whether a turbulence is
fully developed or not is tested with whether the energy spectrum has
a suﬃcient inertial range. Thus, a fully developed turbulence means
that the turbulent ﬂow has a suﬃcient range of the energy spectrum
of E(k) ∝ k
−5/3
.
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10.3. Energy spectrum and energy dissipation 249
10.3.4. Scale of viscous dissipation
Magnitude of the wavenumber associated with dominant dissipation
of energy is estimated according to the following physical reasoning
helped by dimensional arguments. The fully developed turbulence
is regarded as controlled by the viscosity ν (kinematic viscosity)
and the rate of energy dissipation ε per unit mass. It is hypotheti-
cally assumed that viscous dissipation of kinetic energy is predom-
inant at small scales of order l
d
. To compensate the energy loss at
those scales, an energy ﬂow is driven from larger to smaller scales.
At scales in the inertial range, the energy ﬂow from one scale to
smaller scales is almost invariant. Arriving at scales of order l
d
, the
viscous loss becomes dominant. On the other hand, the energy is
injected at a largest scale Λ by some external means with a labo-
ratory scale Λ. This is an idea of energy cascade, envisaged by Kol-
mogorov and Oboukov in the 1940s. Their hypothesis results in the
following dimensional arguments and estimates by scaling. It is called
the similarity hypothesis due to Kolmogorov and Oboukov.
Denoting the dimensions of length and time by L and T, the
dimensions of ν and ε are given by [ε] = L
2
T
−3
and [ν] = L
2
T
−1
,
respectively. This is because, denoting the velocity by U = L/T,
[ε] is given by U
2
/T = L
2
T
−3
(see (10.16)) and [ν] is given by the
relation U/T = [ν]U/L
2
(e.g. see (10.1)), respectively. Now, it can
be immediately shown by these dimensional arguments that the rate
of energy dissipation ε and the viscosity ν determine a characteristic
length scale from their combination which is given by
6
l
d
=

ν
3
ε

1/4
. (10.18)
The l
d
is called Kolmogorov’s dissipation scale. The corresponding
wavenumber is deﬁned by k
d
= 1/l
d
= (ε/ν
3
)
1/4
.
At much higher wavenumbers than k
d
, the spectrum E(k) decays
rapidly because the viscous dissipation is much more eﬀective at
larger k’s. The range k > k
d
is called the dissipation range.
6
Setting ε
α
ν
β
= L
1
, one obtains α = −1/4 and β = 3/4.
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250 Turbulence
10.3.5. Similarity law due to Kolmogorov
and Oboukov
Based on the similarity hypothesis described above, one can derive
an energy spectrum in the inertial range. Suppose that an eddy of
scale l in the inertial range has a velocity of order v
l
. Then, an eddy
turnover time is deﬁned by
τ
l
∼ l/v
l
. (10.19)
The similarity hypothesis implies a constant energy ﬂux ε
l
across
each scale l within the inertial range, given by
ε
l
∼ v
2
l
/τ
l
∼ v
3
l
/l. (10.20)
Setting this to be equal to ε (constant), i.e. ε
l
= ε, we obtain
v
l
∼ (εl)
1/3
. (10.21)
This scaling laws implies that the turbulence ﬁeld is of self-similar
structures between diﬀerent eddy scales l. It is a remarkable property
that the magnitude of velocity characterizing variations over a length
scale l is scaled as l
1/3
, showing a fractionally singular behavior as
l →0.
The energy (per unit mass) of an eddy of scale l is given by
1
2
v
2
l
.
Using the equivalence k = l
−1
, this is regarded as equal to E(k)∆k,
which denotes the amount of energy contained in the wavenumber
range ∆k around k, where ∆k ∼ l
−1
.
7
Thus, we obtain the relation,
E(k)l
−1
≈ v
2
l
∼ (εl)
2/3
, ∴ E(k) ∼ ε
2/3
l
5/3
.
Repacing l with k
−1
, we ﬁnd
E(k) ∼ ε
2/3
k
−5/3
. (10.22)
This is the Kolmogorov–Oboukov’s 5/3-th law.
7
Assuming a power-law sequence of wavenumbers k
0
, k
1
, . . . such as k
n
= a
n
k
0
,
where a and k
0
are constants (say a = 2), we have ∆k = k
n+1
−k
n
= (a −1)k
n
.
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10.4. Vortex structures in turbulence 251
10.4. Vortex structures in turbulence
From the early times of turbulence study by GI Taylor (1938) and
others, it has been recognized that there exist vortical structures
in turbulence. The turbulence spectrum or energy cascade consid-
ered so far are concepts in the space of wavenumbers, while vortices
are structures in real physical space. From the point of view that a
vortical structure is nothing but a dissipative structure, we seek a
deep relation between vortical structures and statistical laws in fully
developed turbulence.
In turbulence, there exists a certain straining mechanism by which
vortex lines are stretched on the average, to be described just below
(Sec. 10.4.1). This is related to the negative value of skewness S[u
x
] of
longitudinal derivative u
x
(Sec. 10.2). Nonzero value of S[u
x
] implies
that the statistics is non-Gaussian, and that there exists structures
in turbulence, often called intermittency. As a slender vortex tube is
stretched, its vorticity increases, and dissipation is enhanced around
it in accordance with (10.15) and the Helmholtz law (iii) of Sec. 7.2.2.
According to the theory of energy cascade of fully developed tur-
bulence, energy is dissipated at scales of smallest eddies of order
l
d
= (ν
3
/ε)
1/4
, the dissipation scale. In computer simulations, the
dissipative structures in turbulence are visualized as ﬁne-scale slen-
der objects with high level of vorticity magnitude.
Figure 10.3 shows a snapshot of a vorticity ﬁeld [YIUIK02]
obtained by a direct numerical simulation (DNS) of an incompressible
turbulence in a periodic box with grid points 2048
3
carried out on the
Earth Simulator (Sec. 8.5), on the basis of the Navier–Stokes equation
(10.1) with an additional term of external random force f under the
incompressible condition (10.2). Figure 10.3(b) is the enlargement of
the central part of Fig. 10.3(a) by eight times. The similarity of struc-
tures between the two ﬁgures implies a self-similarity of an isotropic
homogeneous turbulence in a statistical sense.
10.4.1. Stretching of line-elements
Although turbulence is regarded as a random ﬁeld, it exhibits cer-
tain organized structures of vorticity. It is important that there
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252 Turbulence
Fig. 10.3. High-vorticity isosurfaces obtained by DNS of 2048
3
grid points with
ν = 4.4 ×10
−5
, l
d
= 1.05 ×10
−3
. (a) Length of a side is 2992 l
d
, and (b) 8 times
enlargement of (a), i.e. length of a side is 374 l
d
, the area being 1/64 of that of (a).
The isosurfaces are deﬁned by |ω| = ω + 4σ where σ is the standard deviation
of the distribution of magnitude |ω| [YIUIK02].
exits a fundamental mechanism to stretch vortex-lines in turbulence
ﬁeld. This is closely related to a mechanism by which a small line-
element is stretched always on the average in turbulence. When a
vortex-line is stretched, the vorticity is increased and dissipation is
enhanced by (10.15). Let us ﬁrst consider a mechanism for stretching
line-elements.
Let us consider time evolution of a ﬂuid line-element δx(t) con-
necting two ﬂuid particles A and B at an inﬁnitesimal distance:
δx(t) = x(B, t) −x(A, t).
Its time evolution is described by the following equation for the ith
component,
d
dt
δx
i
= u
i
(B) −u
i
(A) =
∂u
i
∂x
j
δx
j
, (10.23)
by the linear approximation (see (7.13)). The tensor of velocity
derivatives ∂u
i
/∂x
j
can be decomposed always into a symmetric part
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10.4. Vortex structures in turbulence 253
S
ij
and an anti-symmetric part Ω
ij
(Sec. 1.4):
∂u
i
∂x
j
= e
ij
+ Ω
ij
, (10.24)
e
ij
=
1
2
(∂
j
u
i
+ ∂
i
u
j
), Ω
ij
=
1
2
(∂
j
u
i
−∂
i
u
j
), (10.25)
where e
ij
is the rate of strain tensor, while Ω
ij
is the vorticity tensor.
The latter is equivalent to the vorticity vector ω = ∇ v by the
relation Ω
ij
= −
1
2
ε
ijk
ω
k
(Sec. 1.4.3 where Ω
ij
is written as g
ij
). The
(real) tensor e
ij
can be made diagonal by the symmetry e
ij
= e
ji
at each point. In fact, ﬁxing the point x, one can determine three
eigenvalues (E
1
, E
2
, E
3
) of the matrix e
ij
and eigenvectors e
1
, e
2
, e
3
.
By an orthogonal transformation determined by [e
1
, e
2
, e
3
] to the
principal frame, the matrix e
ij
is transformed to a diagonal form
(Sec. 1.4.2): i.e. (e
ij
) = diag(E
1
, E
2
, E
3
).
The velocity gradients ∂u
i
/∂x
j
can be regarded as constant over
a line-element δx(t) if it is suﬃciently short. It can be veriﬁed that
'[δx(t)[
2
` ≥ '[δx(0)[
2
`. In fact, owing to the linearity of Eq. (10.23),
δx(t) is related to δx(0) by a linear relation:
δx
i
(t) = U
ij
δx
j
(0),
where U
ij
is a random tensor (in the turbulent ﬁeld) determined by
the value of ∂
j
u
i
along the trajectory of the line-element, depending
on the initial position and time t, but independent of δx(0). From
the above, we obtain
[δx
i
(t)[
2
= W
jk
δx
j
(0) δx
k
(0), (A)
where W
jk
= U
ij
U
ik
is a real symmetric tensor. Let us denote the
eigenvalues of W
jk
as w
1
, w
2
, w
3
at a ﬁxed time t. Because the
quadratic form W
jk
δx
j
(0)δx
k
(0) is positive deﬁnite by the above
deﬁnition, we should have w
i
> 0 for all i.
Suppose that we take a small sphere of radius a with its center at
a point A at an initial time with its volume (4π/3)a
3
. After a small
time t, it would be deformed to an ellipsoid by the action of turbulent
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254 Turbulence
ﬂow,
8
and the length of the three principal axes would become
√
w
1
a,
√
w
2
a,
√
w
3
a with the volume given by (4π/3)a
3
√
w
1
w
2
w
3
. Owing to
the volume conservation, we must have w
1
w
2
w
3
= 1.
The initial line-element δx(0) can be chosen arbitrarily on the
sphere of radius a. Since the tensor W
jk
of the turbulent ﬁeld is
independent of the choice of δx(0), the values w
1
, w
2
, w
3
are statis-
tically independent of δx(0). In isotropic turbulence, the eigenvectors
associated with w
1
, w
2
, w
3
distribute isotropically. Under these con-
ditions, one can show the property,
' [δx(t)[
2
` =
w
1
+ w
2
+ w
3
3
a
2
≥ (w
1
w
2
w
3
)
1/3
a
2
= a
2
= [δx(0)[
2
,
according to the inequality: (arithmetic mean) ≥ (geometric mean).
Namely, any inﬁnitesimal line-element is always stretched in turbu-
lent ﬁelds in the statistical sense. Similarly, any inﬁnitesimal surface
area is always enlarged in a turbulent ﬁeld in the statistical sense
[Coc69; Ors70].
10.4.2. Negative skewness and enstrophy
enhancement
Vortical structures in turbulent ﬁelds have dual meanings. First,
those signify the structures of velocity ﬁeld. Namely, the ﬁeld may
be irregular, but has some correlation length. Secondly, the vorticity
is related with energy dissipation (see (10.15)). The latter (energy
dissipation) is concerned with the dynamics and time evolution of
the system, whereas the former is concerned with the statistical laws
of spatial distribution. This is the subject at the moment.
8
From (A) and the eigenvalues w
1
, w
2
, w
3
deﬁned below it, we have
|δx
i
(t)|
2
= w
1
(δx
1
(0))
2
+ w
2
(δx
2
(0))
2
+ w
3
(δx
3
(0))
2
. (B)
Since δx(0) was a point on a sphere of radius a, one can write as δx(0) = a(α, β, γ)
where the direction cosines α, β, γ satisfy α
2
+ β
2
+ γ
2
= 1. Analogously,
a point δx(t) = (ξ, η, ζ) on the ellipsoid satisfying the relation (B) can be
expressed by (ξ, η, ζ) = a(
√
w
1
α
√
w
2
β,
√
w
3
γ). Volume of the ellipsoid is given by
(4π/3)(
√
w
1
a)(
√
w
1
a)(
√
w
1
a) = (4π/3)a
3
√
w
1
w
2
w
3
. From the isotropy require-
ment, we have α
2
= β
2
= γ
2
= 1/3.
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10.4. Vortex structures in turbulence 255
Let us consider production of average enstrophy, deﬁned by
1
2
'ω
2
`,
in a homogeneous isotropic turbulence. The equation of vorticity ω
is written as, from (7.3),
∂
t
ω + (v grad)ω = (ω grad)v + ν∇
2
ω, (10.26)
for an incompressible ﬂuid without external force. We used this equa-
tion to obtain the solution of Burgers vortex in Sec. 7.9. Taking a
scalar product of ω with this equation, the left-hand side can be
written as
ω
k
∂
t
ω
k
+ ω
k
(v grad)ω
k
= ∂
t
1
2
ω
2
+ (v grad)
1
2
ω
2
=
D
Dt
1
2
ω
2
,
while the right-hand side is
ω
j
(∂
j
u
i
)ω
i
+ νω
i
∂
j
∂
j
ω
i
=
1
2
ω
j
(∂
j
u
i
+ ∂
i
u
j
)ω
i
+ν∂
j
(ω
i
∂
j
ω
i
) −ν(∂
j
ω
i
)
2
,
since ω
j
(∂
j
u
i
)ω
i
= ω
i
(∂
i
u
j
)ω
j
. Taking an ensemble average, we
obtain
d
dt
1
2
'ω
2
` = 'ω
i
e
ij
ω
j
` −ν

∂ω
i
∂x
j

2
¸
, (10.27)
where e
ij
is deﬁned by (10.25) and the surface integral disappears
when the ensemble average is taken. The right-hand side repre-
sents production or dissipation of the average enstrophy
1
2
'ω
2
`. The
ﬁrst term represents its production associated with the vortex-line
stretching, whereas the second term is its dissipation due to viscos-
ity. The fact that the ﬁrst term expresses enstrophy production is
assured by the property that the skewness of longitudinal velocity
derivative is negative in turbulence, as observed in experiments and
computer simulations.
The skewness of longitudinal velocity derivative is deﬁned by
S
l
=
'(∂u/∂x)
3
`
'(∂u/∂x)
2
`
3/2
,
where the x axis (with u the x component velocity) may be chosen
arbitrarily in the turbulence ﬁeld. From the theory of homogeneous
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256 Turbulence
isotropic turbulence [Bat53; MY71; KD98], one can derive that
'ω
i
e
ij
ω
j
` = −
35
2
S
l

ε
15ν

3/2
. (10.28)
In fully developed turbulence, the skewness S
l
is usually negative
(see Sec. 10.2). Hence, we have
'ω
i
e
ij
ω
j
` > 0.
Thus, it is found that there exists a mechanism in turbulence to
increase the enstrophy on the average which is associated with the
mechanism of vortex-line stretching. Formation of a Burgers vor-
tex studied in Sec. 7.9 is considered to be one of the processes in
turbulence.
10.4.3. Identiﬁcation of vortices in turbulence
In order to identify the vortices in turbulence, a certain criterion is
required, because the vortices in turbulence have no deﬁnite shape.
A simplest way is to apply the div operator to the Navier–Stokes
equation (10.1) under the condition ∂
i
u
i
= 0. Using the relation
∂
j
u
i
= e
ij
+ Ω
ij
of (10.24) and ∂
i
u
i
= 0, we obtain
9
∇
2
p
∗
= −∂
i
u
j
∂
j
u
i
= −(e
ij
+ Ω
ij
)(e
ji
+ Ω
ji
) (10.29)
= −e
ij
e
ji
−Ω
ij
Ω
ji
= −tr¦S
2
+ Ω
2
¦
= −e
ij
e
ji
+ Ω
ij
Ω
ij
= −|S|
2
+|Ω|
2
, (10.30)
where p
∗
= p/ρ
0
, and e
ji
= e
ij
, Ω
ji
= −Ω
ij
are used. We have
introduced the following notations:
|e|
2
≡ e
ij
e
ji
= e
ij
e
ij
, |Ω|
2
≡ Ω
ij
Ω
ij
=
1
2
[ω[
2
.
A vortex may be deﬁned by a domain satisfying the inequality
|Ω|
2
> |S|
2
where the vorticity [ω[ is supposed to be relatively
stronger. In other words, a vortex would be deﬁned by a set of points
characterized by ∇
2
p
∗
> 0. In general, vortex axes are curved lines,
and lower pressure is distributing along them.
9
∂
i
∂
j
(u
i
u
j
) = ∂
i
(u
j
∂
j
u
i
) = ∂
i
u
j
∂
j
u
i
, and Ω
ij
e
ji
+ Ω
ji
e
ij
= Ω
ij
e
ij
−Ω
ij
e
ij
= 0.
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10.4. Vortex structures in turbulence 257
Fig. 10.4. (a) Vortex structure obtained by DNS [TMI97], (b) the circumferential
velocity v
θ
(r) (average by a solid curve; dispersion by vertical bars). The circles
show v
θ
(r) of Burgers vortex.
Figure 10.4(a) shows the vortex structures obtained by a direct
numerical simulation (DNS) of Navier–Stokes equation [TMI97],
visualized by the above method. The diagram (b) on the right
plots the average and dispersion of the circumferential velocity u
θ
(r)
around such vortex structures. The average curve coincides fairly well
with the solution (7.80) of Burgers vortex in Sec. 7.9.
10.4.4. Structure functions
It is natural to say that statistical laws of turbulence are connected
with the distribution of vortex structures in turbulence. This should
provide non-Gaussian statistical laws for velocity correlations at dif-
ferent points in the turbulent ﬁeld.
In order to study a stochastic system of non-Gaussian property,
one of the approaches is to examine structural functions of higher
orders, where the pth order structure function F
p
(s) of longitudinal
velocity diﬀerence ∆v
l
is deﬁned by
F
p
(s) = '(∆v
l
(s))
p
`, (10.31)
the symbol ' ` denoting ensemble average. Suppose that there are
a number of Burgers vortices in turbulence and their axes are being
distributed randomly in the ﬁeld. Choosing two neighboring points x
and x +s in the turbulence ﬁeld, the longitudinal velocity diﬀerence
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258 Turbulence
(with s = [s[) is deﬁned by
∆v
l
(s, x) = (v(x +s) −v(x)) s/s, (10.32)
In the Gaussian system of complete randomness, all the structure
functions of even orders can be represented in terms of the second-
order correlations, while those of odd orders vanish. However, in non-
Gaussian system, one has to know all the structure functions. With
this point of view, the study of statistical properties of turbulence
requires all structure functions.
The velocity ﬁeld of Burgers vortex given in Sec. 7.9 by the expres-
sions (7.76) and (7.80), are reproduced here:
v(x) = (v
r
, v
θ
(r), v
z
) = (−ar, v
θ
(r), 2az), (10.33)
v
θ
(r) =
Γ
2πl
B
1
ˆ r
(1 −e
−ˆ r
2
), (10.34)
where ˆ r = r/l
B
and l
B
=

2ν/a.
At the beginning (Sec. 10.2) of this chapter, we saw that the
ﬂuid turbulence is characterized by a negative value of skewness of
distribution of the longitudinal velocity derivative. The reason why
we consider the Burgers vortices in turbulence is that it is one of
the simplest systems that induces velocity ﬁeld of negative skewness
around it.
This is found by examining the third-order structure function of
velocity ﬁeld around a single Burgers vortex. The structure function
is given by using three eigenvalues σ
1
, σ
2
, σ
3
of the rate of strain
tensor e
ij
=
1
2
(∂
j
v
i
+ ∂
i
v
j
), given in [HK97] as
σ
1
= −a +[e
rθ
[, σ
2
= 2a, σ
3
= −a −[e
rθ
[,
where e
rθ
=
1
2
(v

< 0.
This negativeness is obtained as a result of the combined action of
two basic ﬁelds. In fact, if a = 0, there is no background ﬁeld v
b
= 0
(see (7.75)), and we would have F
3
= 0. On the other hand, if there
is no vortex with Γ = 0 and v
θ
= 0, then we would have F
3
=
(16/35)a
3
s
3
> 0. Thus only the combined ﬁeld of Burgers vortex can
give F
3
< 0, as far as [e
rθ
[ > a > 0.
The scaling property of structure functions F
p
(s) is calculated on
the basis of vortex models in [HK97] and [HDC99]. These give us
fairly good estimates of scaling properties of F
p
(s) in homogeneous
isotropic turbulence.
10.4.5. Structure functions at small s
When the separation distance s is small, the longitudinal structure
functions can be calculated by deﬁning the average ' ` with the space
average,
F
p
(s) = '(∆v
l
(s))
p
`
sp
=
1
4πs
2

= x + s is expressed by
B
ij
:= 'u
i
(x)u
j
(x + s)` = B
ij
(s) which is independent of x by the
assumption of homogeneity. The velocity ﬁeld u
i
(x) is assumed to
satisfy the condition of incompressibility: ∂
i
u
i
(x) = 0.
(i) Suppose that the turbulence is isotropic in addition to homo-
geneity. Show that the second-order tensor B
ij
has the following
form:
B
ij
(s) = G(s)δ
ij
+ F(s)e
i
e
j
, (10.40)
where s = [s[, e
i
= s
i
/s (unit vector in the direction of s), and
F(s) and G(s) are scalar functions.
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10.5. Problems 261
(ii) For turbulence of an incompressible ﬂuid, show that the follow-
ing relation must be satisﬁed:
2F(s) + sF

. (10.55)
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Chapter 11
Superﬂuid and quantized
circulation
At very low temperatures close to absolute zero, quantum eﬀects
begin to acquire primary importance in the properties of ﬂuids.
It is well-known that helium becomes a liquid phase below the
(critical) temperature T
c
= 4.22 K (under atmospheric pressure),
and superﬂuid properties appear below T
λ
= 2.172 K (discovered by
P. L. Kapitza, 1938).
1
Recently, there has been dramatic improvement in the Bose–
Einstein condensation of (magnetically) trapped alkali-atomic gases
at ultra-low temperatures. Such an atomic-gas Bose–Einstein con-
densation diﬀers from the liquid-helium condensate in several ways.
An example of this is that condensates of alkali-atomic gases are
dilute. As a result, at low temperatures, the Gross–Pitaevskii equa-
tion (11.23) below gives an extremely precise description of the
atomic condensate, and its dynamics is described by potential ﬂows
of an ideal ﬂuid with a uniform (vanishing) entropy.
In traditional ﬂuid dynamics, the ideal ﬂuid is a virtual idealized
ﬂuid which is characterized by vanishing transport coeﬃcients such
as viscosity and thermal conductivity. The superﬂuid at T = 0
◦
K is
a real ideal ﬂuid which supports only potential ﬂows of zero entropy.
However, it is remarkable that it can support quantized circulations
1
From T
λ
down to T = 0
◦
K, the liquid is called helium II. Under saturated vapor
pressure, liquid helium is an ordinary classical viscous ﬂuid called helium I from
T
c
down to T
λ
. The theory of superﬂuidity was developed by L. D. Landau (1941).
263
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264 Superﬂuid and quantized circulation
as well at excited states. The purpose of this chapter is to introduce
such ideas of superﬂuid ﬂows.
2
11.1. Two-ﬂuid model
At the temperature of absolute zero, helium II is supposed to show
superﬂuidity of a Bose liquid obeying the Bose–Einstein statistics,
3
and it ﬂows without viscosity in narrow capillaries or along a solid
surface. Besides the absense of viscosity, the superﬂuid ﬂow has two
other important properties. Namely, the ﬂow is always a potential
ﬂow. Hence, the velocity v
s
of such a superﬂuid ﬂow has a velocity
potential Φ deﬁned as
v
s
= grad Φ. (11.1)
In addition, its entropy is zero.
At temperatures other than zero, helium II behaves as if it were
a mixture of two diﬀerent liquids. One of them is a superﬂuid, and
moves with zero viscosity. The other is a normal ﬂuid with viscosity.
4
Helium II is regarded as a mixture of normal ﬂuid and superﬂuid with
total density ρ:
ρ = ρ
s
+ρ
n
. (11.2)
The densities of the superﬂuid ρ
s
and the normal ﬂuid ρ
n
are known
as a function of temperature and pressure. At T = T
λ
, ρ
n
= ρ and
2
For the details, see [LL87] and [Don91].
3
A Bose particle (called a boson) is characterized by a particle with an integer
spin. The wavefunction for a system of same bosons is symmetric with respect to
exchange of arbitrary two particles.This is in contrast with the fermions of semi-
integer spins whose wavefunction is anti-symmetric with respect to the exchange.
Any number of bosons can occupy the same state, unlike the fermions.
4
The liquid helium is regarded as a mixture of normal ﬂuid and superﬂuid parts.
This is no more than a convenient description of the phenomena in a quan-
tum ﬂuid. Just like any description of quantum phenomena in classical terms,
it falls short of adequacy. In reality, it should be said that a quantum ﬂuid
such as helium II can execute two motions at once, each of which involves its
own diﬀerent eﬀective mass. One of these motions is normal, while the other is
superﬂuid-ﬂow.
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11.1. Two-ﬂuid model 265
ρ
s
= 0. At T = 0, ρ
n
= 0 and ρ
s
= ρ. The total density ρ is nearly
constant in the helium II temperature range. At temperatures lower
than 1K, ρ ≈ ρ
s
.
The ﬂow of ﬂuid is characterized by the normal velocity v
n
and
the superﬂuid velocity v
s
. The ﬂow of the superﬂuid is irrotational:
curl v
s
= 0, (11.3)
which must hold throughout the volume of the ﬂuid. Hence, v
s
has
a velocity potential Φ deﬁned by (11.1).
Total mass ﬂux is deﬁned by the sum of two components:
j = ρ
s
v
s
+ρ
n
v
n
. (11.4)
The total density ρ and mass ﬂux j must satisfy the continuity
equation:
∂
t
ρ + div j = 0. (11.5)
The law of conservation of momentum is given by
∂
t
j
i
+∂
k
P
ik
= 0, (11.6)
where the momentum ﬂux density tensor P
ik
is deﬁned by
P
ik
= pδ
ik
+ρ
s
(v
s
)
i
(v
s
)
k
+ρ
n
(v
n
)
i
(v
n
)
k
. (11.7)
See the expressions (3.21) (with f
i
= 0) and (3.22) in the case of an
ideal ﬂuid.
The superﬂuid part involves no heat transfer, and therefore no
entropy transfer. The entropy of superﬂuid is regarded as zero.
The heat transfer in helium II is caused only by the normal ﬂuid.
The entropy ﬂux density is given by the product ρsv
n
where s is the
entropy per unit mass and ρs the entropy per unit volume. Neglect-
ing the dissipative processes, the entropy of the ﬂuid is conserved,
which is written down as
∂
t
(ρs) + div(ρsv
n
) = 0. (11.8)
The heat ﬂux density q is expressed as
q = ρTsv
n
. (11.9)
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266 Superﬂuid and quantized circulation
Equations (11.5)–(11.8) are supplemented by the equation for the
superﬂuid part v
s
. The equation must be such that the superﬂuid
ﬂow is irrotational at all times, which is given by Euler’s equation of
motion for irrotational ﬂows:
∂
t
v
s
+ grad

1
2
v
2
s
+µ

= 0, (11.10)
where µ is a scalar function (corresponding to the chemical potential
in thermodynamics). In Sec. 5.4, we obtained Eq. (5.24) for an irrota-
tional ﬂow, where there were scalar functions of enthalpy h and force
potential χ in place of µ. Using the expression (11.1), Eq. (11.10) is
rewritten as
grad ∂
t
Φ + grad

1
2
v
2
s
+µ

= 0.
This results in the following ﬁrst integral of motion (see (5.28)):
∂
t
Φ +
1
2
(∇Φ)
2
+µ = const. (11.11)
11.2. Quantum mechanical description of
superﬂuid ﬂows
11.2.1. Bose gas
At present there is no truly fundamental microscopic picture of super-
ﬂuidity of helium II. However, there are some ideas about superﬂu-
idity based on quantum mechanics.
4
He atoms are bosons and hence we try to draw an analogy
between helium II and an ideal Bose gas (see the footnote in Sec. 11.1
for Bose–Einstein statistics). An ideal Bose gas begins to condense
into the lowest ground energy level below the critical temperature
T
c
. When T = 0
◦
K, all particles of a number N are in the ground
state. At a ﬁnite temperature T (< T
c
), the fraction of condensation
to the ground state is given by the form, N
0
(T) = N [1 −(T/T
c
)
3/2
].
The condensed phase consists of N
0
(T) particles which are in the
zero-momentum state, while the uncondensed particles of the num-
ber N − N
0
= N(T/T
c
)
2/3
are distributing over excited states. It is
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11.2. Quantum mechanical description of superﬂuid ﬂows 267
tempting to associate the superﬂuid with particles in the ground
state, while a system of excitations is regarded as a normal ﬂuid.
The normal ﬂow (ﬂow of the normal ﬂuid) is actually the ﬂow of an
excitation gas.
5
11.2.2. Madelung transformation and hydrodynamic
representation
Superﬂuid ﬂows can be described by Schr¨ odinger’s equation
(Sec. 12.1) in quantum mechanics. Schr¨ odinger’s equation for the
wave function ψ(r, t) can be transformed to the forms familiar in the
ﬂuid mechanics (Madelung, 1927) by expressing ψ(r, t) in terms of
its amplitude |ψ| = A(r, t)
6
and its phase arg(ψ) = ϕ(r, t) where A
and ϕ are real functions of r and t. It is remarkable that the velocity
u can be connected with the gradient of phase ϕ.
Schr¨ odinger’s wave equation for a single particle of mass m in a
ﬁxed potential ﬁeld V (r) is given by
i ∂ψ/∂t = −(
2
/2m)∇
2
ψ +V ψ, (11.12)
where = h/2π and h is the Planck constant (h = 6.62×10
−34
J · s).
Substituting
ψ = A exp[iϕ] (11.13)
into (11.12), dividing it into real and imaginary parts, we obtain
∂
t
ϕ −

2
2m
∇
2
A
A
+

2
2m
(∇ϕ)
2
+V = 0, (11.14)
2∂
t
A+ 2

m
∇A · ∇ϕ +
A
m
∇
2
ϕ = 0. (11.15)
The mass probability density is deﬁned by ρ
ρ = mψψ
∗
= mA
2
. (11.16)
5
We may recall that the collective thermal motion of atoms in a quantum ﬂuid can
be regarded as a system of excitations. The excitations behave like quasi-particles
with carrying deﬁnite momenta and energies.
6
A
2
= |ψ|
2
is understood as the probability density for the particle to be observed.
See standard textbooks, e.g. [LL77].
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268 Superﬂuid and quantized circulation
where ψ
∗
is the complex conjugate of ψ. The mass current is
deﬁned by
j = (/2i)

ψ
∗
∇ψ −ψ∇ψ
∗

= A
2
∇ϕ. (11.17)
From the last two equations, we can derive a velocity u by requiring
that the mass ﬂux must have the form j = ρu. Thus, we obtain
u =

m
∇ϕ, (11.18)
which has the expected form (11.1) with Φ = (/m)ϕ.
Multiplying (11.15) by mA and using (11.16), we ﬁnd the conti-
nuity equation,
∂
t
ρ +∇· (ρu) = 0, u = (/m)∇ϕ. (11.19)
Using Φ = (/m)ϕ, Eq. (11.14) becomes
∂
t
Φ +
1
2
(∇Φ)
2
+B +V/m = 0, (11.20)
where B = −(
2
/2m
2
)(∇
2
A/A). This is seen to be equivalent to the
integral (11.11) if B +V/m is replaced by the scalar function µ.
Thus, it is found that the Schr¨ odinger equation (11.12) is equiv-
alent to the system of equations (11.4), (11.5) and (11.10) for the
superﬂuid ﬂow ρ = ρ
s
with ρ
n
= 0. By normalization of the wave-
function given by

|ψ|
2
d
3
r = 1, (11.21)
we have the relation

ρd
3
r = m.
11.2.3. Gross–Pitaevskii equation
Instead of a single particle, we now consider an assembly of N
0
iden-
tical bosons of mass m in a potential ﬁeld W(r) in order to represent
the macroscopic condensate of bosons (macroscopic Bose–Einstein
condensation). If these bosons do not interact, the wavefunction
ψ(r, t) of the system would be given by a symmetrized product of N
0
one-particle wavefunctions with the normalization condition (11.21)
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11.3. Quantized vortices 269
replaced by

|ψ|
2
d
3
r = N
0
, (11.22)
or

ρd
3
r = ρ V, where V is the volume of the system, and ρV is
deﬁned by mN
0
.
The condensate phase has a typical interatomic potential includ-
ing a strong repulsive potential, which is expressed by a short-range
repulsive potential V
r
(x − x

). In the Schr¨odinger equation, the fol-
lowing potential,

V
r
(x −x

)|ψ(x

)|
2
d
3
x

,
is added to V . This increases, as the density of neighboring bosons
increases. A simplest case is to express V
r
by a delta function
V
r
(x −x

m
∂
t
ϕ +
1
2
u
2
+V/m +B +
p
ρ
= 0. (11.24)
This equation diﬀers from (11.20) principally by the additional term
p/ρ = V
0
|ψ|
2
/m = (V
0
/m
2
)ρ, corresponding to a barotropic gas pres-
sure p = (V
0
/m
2
)ρ
2
.
11.3. Quantized vortices
Vortices in the superﬂuid are considered to take the form of a (hollow)
ﬁlament with a core of atomic dimensions, something like a vortex-
line (Feynman 1955). The well-known invariant called the hydrody-
namic circulation is quantized; the quantum of circulation is h/m.
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270 Superﬂuid and quantized circulation
In the case of cylindrical symmetry, the angular momentum per par-
ticle is (Onsager 1949). With this structure, multiple connectivity
in the liquid arises because the superﬂuid is somehow excluded from
the core and circulates about the core.
11.3.1. Quantized circulation
The ﬂuid considered is a quantum ﬂuid described by the wavefunc-
tion ψ, and one would expect some diﬀerences associated with the
quantum character. Suppose that the ﬂuid is conﬁned to a multiply-
connected region. Then, around any closed contour C (not reducible
to a point by a continuous deformation), the phase ϕ of (11.13) can
change by a multiple of 2π. This is written as

C
u · dl = n
h
m
(n = 0, ±1, ±2, . . .). (11.25)
Thus, it has been shown that the circulation Γ around any closed
contour can have only integer multiples of a unit value h/m. This
coincides with the above statement of Onsager.
Suppose that a steady vortex is centered at r = 0. By (11.25), we
have the circumferential velocity of magnitude,
u
θ
=

W/V
0
→1, (11.30)
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272 Superﬂuid and quantized circulation
Fig. 11.1. A solution of a hollow vortex: The vertical axis is A
2
, proportional to
density ρ [GP58; KP66].
at large distances as ζ →∞.
8
Then, from (11.24), we get the follow-
ing equation,
d
2
dζ
2
A+
1
ζ
d
dζ
A−
1
ζ
2
A+A−A
3
= 0. (11.31)
It is readily seen that there is a solution A(ζ) satisfying this equa-
tion. For small distances ζ from the axis, the solution A(ζ) varies
linearly as ζ, and at large distances (ζ 1), A = 1 −
1
2
ζ
−2
+ · · ·
(Problem 11.2). To obtain the whole solution A(ζ), Eq. (11.31) must
be solved numerically (Fig. 11.1). This was carried out by [GP58]
and [KP66].
The “coherence” here refers to the condensate wavefunction
falling to zero inside the quantum liquid, which is the result of inter-
particle interactions. In this model, the vortex is characterized by a
line-node (zero-line) in the wavefunction, around which circulation
can take a value of an integer multiple of h/m.
8
There are N
0
condensate particles in volume V (see Sec. 11.2.3), and the density
is given by ρ
∞
= mN
0
/V = mW/V
0
by the normalization (11.30).
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11.4. Bose–Einstein Condensation (BEC) 273
11.4. Bose–Einstein Condensation (BEC)
11.4.1. BEC in dilute alkali-atomic gases
Recently, there has been a dramatic improvement in Bose–Einstein
condensation of magnetically trapped alkali-atomic gases at ultra-
low temperatures [PS02]. Such atomic gas Bose–Einstein condensa-
tion diﬀers from liquid helium condensates in several ways. First, the
condensates of alkali-atomic gases are dilute, having the mean par-
ticle density n with na
3
1 for interaction length a. As a result, at
low temperatures, the GP equation (11.23) gives an extremely precise
description of atomic condensate and their dynamics. In superﬂuid
4
He, relatively high density and strong repulsive interactions make
the analysis more complicated. Furthermore, because of the relatively
strong interactions, the condensate fraction in bulk superﬂuid
4
He is
only about 10% of the total particles, even at zero temperature. In
contrast, almost all atoms participate in the condensate in an atomic-
gas BEC. In 1995, BEC was ﬁrst realized for a cluster of dilute Rb
and Na atoms. Since then, with many alkali-atomic gases, BEC was
realized experimentally. After 1998, BEC was conﬁrmed also for H
and He.
BEC is diﬀerent from the liqueﬁed gas in the normal phase
change at ordinary temperatures. The liquefaction requires interac-
tion between atoms, while no interaction is required with BEC. More-
over, all the atoms composing the BEC quantum state are identical
and cannot be distinguished, and represented by a single wavefunc-
tion of a macroscopic scale (of 1 mm, say) for a macroscopic number
of atoms.
However, the number density must be suﬃciently low, i.e. dilute.
Usually, atomic gases are liqueﬁed when the temperature is lowered.
Atoms in a condensate state as in liquid or solid are bounded by
a certain interatomic attractive potential such as the van der Waals
potential. In other words, atoms must release the bounding energy to
some agent. This is carried out by a three-body collision (or higher
order) in a gas away from solid walls. This is not the case in the
two-body collision in which energy and momentum are conserved by
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274 Superﬂuid and quantized circulation
two particles only and any bound state cannot be formed. When the
number density n is low enough, two-body collision becomes domi-
nant since probability of m-body collision is proportional to n
m
.
Realization of BEC of dilute alkali-atomic gases is the most
remarkable experimental achievement in the last decades at the end
of 20th century. For experimental realization of BEC, three tech-
niques must be combined: (a) laser cooling, (b) magnetic trap of gas
particles in a bounded space, and (c) evaporation cooling.
11.4.2. Vortex dynamics in rotating BEC condensates
Madison et al. [MCBD01] reported the observation of nonlinear
dynamics of quantum vortices such as vortex nucleation and lattice
formation in a rotating condensate trapped by a potential, together
Fig. 11.2. Time evolution of the density of a rotating two-dimensional Bose
condensate from the initial circular state, governed by the GP equation in a
rotating frame with angular velocity Ω: (i −γ) ∂
t
ψ = −(
2
/2m)∇
2
ψ+(V
0
|ψ|
2
+
V − ΩL
z
− µ)ψ, where γ is a damping constant, µ a chemical potential and L
z
the z-angular momentum [Tsu03].
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11.5. Problems 275
with detailed observation of time evolution of condensate deforma-
tion. Such a direct observation of vortex phenomena has been made
possible only in the BEC of alkali-atomic gases. Computer simulation
[TKA03] of corresponding vortex dynamics was carried out success-
fully by numerically solving an extended form of the GP equation
(Fig. 11.2). The physics of quantized vortices has a novel feature.
Since the density is dilute, the relatively large coherence length a
makes it possible to directly visualize the quantized vortices by using
optical means. Secondly, because the order of coherence length a is
close to the size of the condensate, the vortex dynamics is closely
connected with the collective motion of the condensate density.
This opens a new area of vortex dynamics of an ideal ﬂuid.
11.5. Problems
Problem 11.1 Bessel’s diﬀerential equation
Bessel’s diﬀerential equation of the nth order is given by
d
2
dr
2
A +
1
r
d
dr
A+

1 −
n
2
r
2

A = 0 , (11.32)
where n = 0, 1, 2, . . . (zero or positive integers). Show that this has a
following power series solution A(r) = J
n
(r):
J
n
(r) =

r
2

n
∞
¸
k=0
(−1)
k
1
k! (n +k)!

r
2

2k
=

r
2

n
¸
1
n!
−
1
1!(n + 1)!

r
2

2
+
1
2!(n + 2)!

r
2

4
−· · ·

,
(11.33)
where 0! := 1, 1! := 1, and k! := 1 · 2 · · · · · k for an integer k > 0.
[Note: Equation (11.29) corresponds to n = 1.]
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276 Superﬂuid and quantized circulation
Problem 11.2 Axisymmetric Gross–Pitaevskii equation
Axisymmetric Gross–Pitaevskii equation is given by (11.31):
d
2
dζ
2
A+
1
ζ
d
dζ
A−
1
ζ
2
A+A−A
3
= 0. (11.31)
Show that, (i) for small ζ, A(ζ) = c ζ +O(ζ
3
) (c: a constant); (ii) for
large ζ (1), A = 1 −
1
2
ζ
−2
+· · · .
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Chapter 12
Gauge theory of ideal ﬂuid ﬂows
Fluid mechanics is a ﬁeld theory in Newtonian mechanics, i.e. a ﬁeld
theory of mass ﬂows subject to Galilean transformation. In the gauge
theory of particle physics and the theory of relativity, a guiding prin-
ciple is that laws of physics should be expressed in a form that is
independent of any particular coordinate system. It is well known
that there are various similarities between ﬂuid mechanics and elec-
tromagnetism. Gauge theory provides us a basis to reﬂect on this
property.
There are obvious diﬀerences between the ﬁeld of ﬂuid ﬂows
and other ﬁelds of electromagnetism, quantum physics, or particle
physics. Firstly, the ﬁeld of ﬂuid ﬂow is nonquantum. However this
causes no problem since the gauge principle is independent of the
quantization principle. In addition, the ﬂuid ﬂow is subject to the
Galilean transformation instead of the Lorentz transformation in
the ﬁeld theory. This is not an obstacle because the former (Galilean)
is a limiting transformation of the latter (Lorentz) as the ratio of
a ﬂow velocity to the light speed tends to an inﬁnitesimally small
value. Thirdly, the gauge principle in the particle physics requires
the system to have a symmetry, i.e. the gauge invariance, which is an
invariance with respect to a certain group of transformations. The
symmetry groups (i.e. gauge groups) are diﬀerent between diﬀerent
physical systems. This will be considered below in detail.
In the formulation of ﬂuid ﬂows, we seek a scenario which
has a formal equivalence with the gauge theory in physics. An
essential building block of the gauge theory is the covariant
277
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278 Gauge theory of ideal ﬂuid ﬂows
derivative. The convective derivative D/Dt, i.e. the Lagrange deriva-
tive, in ﬂuid mechanics can be identiﬁed as the covariant derivative
in the framework of gauge theory. Based on this, we deﬁne appropri-
ate Lagrangian functions for motions of an ideal ﬂuid below. Euler’s
equation of motion can be derived from the variational principle. In
addition, the continuity equation and entropy equation are derived
simultaneously.
12.1. Backgrounds of the theory
We review the background of gauge theory ﬁrst and illustrate the
scenario of the gauge principle in a physical system.
12.1.1. Gauge invariances
In the theory of electromagnetism, it is well known that there is an
invariance under a gauge transformation of electromagnetic poten-
tials consisting of a scalar potential φ and a vector potential A. The
electric ﬁeld E and magnetic ﬁeld M are represented as
E = −∇φ −
1
c
∂A
∂t
, B = ∇A, (12.1)
where c is the light velocity. The idea is that the ﬁelds E and M
are unchanged by the following transformation: (φ, A) → (φ

, A

),
where φ

= φ −c
−1
∂
t
f, A

= A+∇f with f(x, t) being an arbitrary
diﬀerentiable scalar function of position x and time t. It is readily
seen that
B

≡ ∇A

= ∇A = B (since ∇∇f = 0),
E

≡ −c
−1
∂
t
A

−∇φ

= −c
−1
∂
t
A−∇φ = E.
In ﬂuid mechanics, it is interesting to ﬁnd that there is an analogous
invariance under (gauge) transformation of a velocity potential Φ of
irrotational ﬂows of an ideal ﬂuid, where the velocity is represented
as v = ∇Φ. This was noted brieﬂy as a gauge invariance in the
footnote in Sec. 5.4. There, a potential ﬂow of a homentropic ﬂuid
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12.1. Backgrounds of the theory 279
had an integral of motion (5.28) expressed as
∂
t
Φ +
1
2
v
2
+ H = const., (12.2)
where H denotes the summation of the enthalpy h and a force poten-
tial χ (H = h+χ). It is evident that all of the velocity v, the integral
(12.2) and the equation of motion (5.24) are unchanged by the trans-
formations: Φ → Φ + f(t) and H → H − ∂
t
f(t) with an arbitrary
scalar function f(t). This case is simpler because only a scalar poten-
tial Φ and a scalar function (of t only) are concerned.
In short, there exists arbitrariness in the expressions of physical
ﬁelds in terms of potentials in both systems.
12.1.2. Review of the invariance in quantum
mechanics
First we highlight a basic property in quantum mechanics. Namely,
for a charged particle of mass m in electromagnetic ﬁelds,
Schr¨ odinger’s equation and the electromagnetic ﬁelds are invariant
with respect to a gauge transformation. This is as follows.
In the absence of electromagnetic ﬁelds, Schr¨odinger’s equation
for a wave function ψ of a free particle m is written as
1
S[ψ] ≡ i∂
t
ψ −
1
2m
p
2
k
ψ = 0,
where p
k
= −i∂
k
is the momentum operator (∂
k
= ∂/∂x
k
for k =
1, 2, 3).
In the presence of electromagnetic ﬁelds, Schr¨odinger’s equation
for a particle with an electric charge e is obtained by the following
transformation from the above equation:
∂
t
→∂
t
+(e/i)A
0
, ∂
k
→∂
k
+(e/ic)A
k
, (k = 1, 2, 3), (12.3)
1
This is an equation for a free particle since the potential V is set to zero in
(11.12).
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280 Gauge theory of ideal ﬂuid ﬂows
where (A
µ
) = (A
0
, A
1
, A
2
, A
3
) = (A
0
, A) = (−φ, A) is the four-
vector potential.
2
Thus, we obtain the equation with electromagnetic
ﬁelds:
S
A
[ψ] ≡ i∂
t
ψ −eφψ −
1
2m

(−i)

∂
k
+
e
ic
A
k

2
ψ = 0. (12.4)
From the four-vector potential A
µ
= (−φ, A), we obtain the electric
ﬁeld E and magnetic ﬁeld B represented by (12.1).
Suppose that a wave function ψ(x
µ
) satisﬁes the equation
S
A
[ψ] = 0 of (12.4). Consider the following set of transformations
of ψ(x
µ
) and A
µ
:
ψ

(x
µ
) satisﬁes the Schr¨odinger equation S
A
[ψ] = 0,
obtained by replacing the potential A
µ
with the transformed one A

µ
.
In addition, if
α ≡ (∇
2
−c
−2
∂
2
t
)α = 0
is satisﬁed, then the electromagnetic ﬁelds E and B are invariant,
and the probability density is also invariant: [ψ

(x
µ
)[
2
= [ψ(x
µ
)[
2
.
This is the gauge invariance of the system of an electric charge in
electromagnetic ﬁelds.
In summary, the Schr¨ odinger equation (12.4) and the electromag-
netic ﬁelds E and B are invariant with respect to the gauge trans-
formations, (12.5) and (12.6). This system is said to have a gauge
symmetry. Namely, the system has a certain kind of freedom. This
freedom (or symmetry) allows us to formulate the following gauge
principle.
2
A point in the space-time frame of reference is expressed by a four-component
vector, (x
µ
) = (x
0
, x
1
, x
2
, x
3
) with x
0
= ct (upper indices). Corresponding form
of the electromagnetic potential is represented by a four-covector (lower indices)
A
µ
= (−φ, A
k
) with φ the electric potential and A = (A
k
) the magnetic three-
vector potential (k = 1, 2, 3). Transformation between a vector A
ν
and a covec-
tor A
µ
is given by the rule A
µ
= g
µν
A
ν
with the metric tensor g
µν
deﬁned in
Appendix F.
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12.1. Backgrounds of the theory 281
12.1.3. Brief scenario of gauge principle
In the gauge theory of particle physics,
3
a free-particle Lagrangian
L
free
is ﬁrst deﬁned for a particle with an electric charge. Next, a
gauge principle is applied to the Lagrangian, requiring it to have a
gauge invariance.
Gauge Transformation: Suppose that a Lagrangian L
free
[ψ] is deﬁned
for the wave function ψ of a free charged particle. Let us consider
the following (gauge) transformation:
ψ →e
iα
ψ.
If L
free
is invariant with this transformation when α is a constant, it
is said that L
free
has a global gauge invariance. In spite of this, L
free
may not be invariant for a function α = α(x). Then, it is said that
L
free
is not gauge-invariant locally.
In order to acquire local gauge invariance, let us introduce a new
ﬁeld. Owing to this new ﬁeld, if local gauge invariance is recovered,
then the new ﬁeld is called a gauge ﬁeld.
In the previous section, it was found that local gauge invariance
was acquired by replacing ∂
µ
with
∇
µ
= ∂
µ
+/
µ
(12.7)
(see (12.3)), where /
µ
= (e/ic)A
µ
, and A
µ
(x) is the electromagnetic
potential, termed as a connection form in mathematics. The operator
∇
µ
is called the covariant derivative.
Thus, when the original Lagrangian is not locally gauge invariant,
the principle of local gauge invariance requires a new gauge ﬁeld to
be introduced in order to acquire local gauge invariance, and the
Lagrangian is to be altered by replacing the partial derivative with
the covariant derivative including a gauge ﬁeld. This is the Weyl’s
gauge principle. Electromagnetic ﬁeld is a typical example of such a
kind of gauge ﬁeld.
In mathematical terms, suppose that we have a group ( of trans-
formations and an element g(x) ∈ (, and that the wave function is
transformed as ψ

= g(x)ψ. In the previous example, g(x) = e
iα(x)
3
[Wnb95] or [Fr97].
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282 Gauge theory of ideal ﬂuid ﬂows
and the group is ( = U(1).
4
An important point of introducing the
gauge ﬁeld A is to deﬁne a covariant derivative ∇ = ∂ +A as a gen-
eralization of the derivative ∂ that transforms as g∇ = g(∂ + A) =
(∂

+ A

)g, so that we obtain
∇

ψ

= (∂

+ A

)g(x)ψ = g(∂ + A)ψ = g∇ψ. (12.8)
Thus, ∇ψ transforms in the same way as ψ itself.
In dynamical systems which evolve with time t, such as ﬂuid ﬂows
(in an Euclidean space), the replacement is to be made only for the
t derivative: ∂
t
→ ∇
t
= ∂
t
+ A(x). This is considered in the next
section.
12.2. Mechanical system
To start with, we review the variational formulation in Newtonian
mechanics of a system of point masses, aiming at presenting a new
formulation later. Then we consider an invariance property with
respect to certain group of transformations, i.e. a symmetry of the
mechanical system.
12.2.1. System of n point masses
Suppose that we have a system of n point masses m
j
(j = 1, . . . , n)
whose positions x
j
are denoted by q = (q
i
), where
x
1
= (q
1
, q
2
, q
3
), . . . , x
n
= (q
3n−2
, q
3n−1
, q
3n
).
Their velocities v
j
= (v
1
j
, v
2
j
, v
3
j
) are written as (q
3j−2
t
, q
3j−1
t
, q
3j
t
).
Usually, a Lagrangian function L is deﬁned by (Kinetic energy) −
(Potential energy) for such a mechanical system. However, we start
here with the following form of Lagrangian:
L = L[q, q
t
], (12.9)
which depends on the coordinates q = q(t) and the velocities q
t
=
∂
t
q(t) = (q
i
t
) for i = 1, 2, . . . , 3n (∂
t
= d/dt). The Lagrangian L
describes a dynamical system of 3n degrees of freedom.
4
Unitary group U(1) is the group of complex numbers z = e
iθ
of absolute value 1.
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12.2. Mechanical system 283
For derivation of the equation of motion, we rely on the varia-
tional principle as follows. Given two end times t
1
and t
2
, the action
functional I is deﬁned by I =

. (12.11)
Substituting the expression (12.11) into (12.10), the last term can
be integrated, giving the diﬀerence of values at t
1
and t
2
, both of
which vanish by the imposed conditions: ξ(t
1
) = ξ(t
2
) = 0. Thus, we
obtain
δI =

t
2
t
1

∂L
∂q
i
−
d
dt

∂L
∂q
i
t

δq
i
dt = 0.
The variational principle requires that this must be valid against
any variation δq
i
satisfying δq
i
(t
1
) = δq
i
(t
2
) = 0. This results in the
following Euler–Lagrange equation:
d
dt

. (12.14)
When the displacement ξ is a constant vector for all x
j
(j = 1, . . . , n)
like in the present case, the transformation is called global. Requir-
ing that the Lagrangian is invariant under this transformation, i.e.
δL = 0, we obtain
0 =
d
dt

∂L
∂q
i
t
δq
i

= ξ
k
3
¸
k=1
d
dt

n
¸
j=1
∂L
∂v
k
j

.
5
This is equivalent to a shift of the coordinate origin by −ξ. This global gauge
transformation is diﬀerent from the variational principle considered in the previ-
ous section.
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12.3. Fluid as a continuous ﬁeld of mass 285
Since ξ
k
(k = 1, 2, 3) are arbitrary independent constants, the fol-
lowing must hold:
n
¸
j=1
∂L
∂v
1
j
= const.,
n
¸
j=1
∂L
∂v
2
j
= const.,
n
¸
j=1
∂L
∂v
3
j
= const.
Thus the three components of the total momentum are conserved.
This is the Noether’s theorem for the global invariance of the
Lagrangian L of (12.9). It is well known that the Newton’s equation
of motion is invariant with respect to Galilean transformation, i.e. a
transformation between two inertial frames of reference in which one
frame is moving with a constant velocity U relative to the other. The
Galilean transformation is a sequence of global translational gauge
transformations with respect to the time parameter t. Global invari-
ance of Lagrangian with respect to translational transformations is
associated with the homogeneity of space.
Lagrangian of (12.9) has usually another invariance associated
with the isotropy of space. Namely, the mechanical property of the
system is unchanged when it is rotated as a whole in space. This
is a rotational transformation, i.e. a position vector of every parti-
cle in the system is rotated by the same angle with respect to the
space. The global symmetry with respect to the rotational trans-
formations results in the conservation of total angular momentum
([LL76], Secs. 7 and 9).
It is to be noted that the gauge symmetry of the Schr¨odinger
system considered in Sec. 12.1.2 was local. The next section investi-
gates extension of the above global symmetry to local symmetry for
a continuous mechanical system.
12.3. Fluid as a continuous ﬁeld of mass
From now on, we consider ﬂuid ﬂows and try to formulate the ﬂow
ﬁeld on the basis of the gauge principle. First, we investigate how the
Lagrangian of the form (12.9), or (12.13), of discrete systems must
be modiﬁed for a system of ﬂuid ﬂows characterized by continuous
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286 Gauge theory of ideal ﬂuid ﬂows
distribution of mass. According to the principle of gauge invariance,
we consider gauge transformations, which are both global and local.
Concept of local transformation is a generalization of the global
transformation. When we consider local gauge transformation, the
physical system under consideration must be modiﬁed so as to allow
us to consider a continuous ﬁeld by extending the original discrete
mechanical system. We replace the discrete variables q
i
by continuous
parameters a = (a
1
, a
2
, a
3
) to represent continuous distribution of
particles in a subspace M of three-dimensional Euclidean space E
3
.
Spatial position x = (x
1
, x
2
, x
3
) of each massive particle of the name
tag a (Lagrange parameter) is denoted by x = x
a
(t) ≡ X(a, t), a
function of a as well as the time t. Conversely, the particle occupying
the point x at a time t is denoted by a(x, t).
12.3.1. Global invariance extended to a ﬂuid
Now, we consider a continuous distribution of mass (i.e. ﬂuid) and
its motion. The Lagrangian (12.9) must be modiﬁed to the following
integral form (instead of summation):
L =

∂Λ
∂v
d
3
x = 0.
This states the conservation of total momentum deﬁned by

(∂Λ/∂v)d
3
x. The same result for the global transformation (ξ =
const. and δv = ∂
t
ξ = 0) can be obtained directly from (12.15) since
δL =

ξ
∂Λ
∂x
d
3
x = ξ

∂
t

∂Λ
∂v

d
3
x = 0, (12.19)
by the above Euler–Lagrange equation. In the local transformation,
however, the variation ﬁeld ξ depends on the time t and space point
x, and the variation δL =

δΛd
3
x does not vanish in general.
12.3.2. Covariant derivative
According to the gauge principle (Sec. 12.1.3), nonvanishing of δL
is understood as meaning that a new ﬁeld G must be taken into
account in order to achieve local gauge invariance of the Lagrangian
L. To that end, we try to replace the partial time derivative ∂
t
by a
covariant derivative D
t
, where the derivative D
t
is deﬁned by
D
t
= ∂
t
+ G, (12.20)
with G being a gauge ﬁeld (an operator).
In dynamical systems like the present case, the time derivative is
the primary object to be considered in the analysis of local gauge
transformation. (The invariance property noted in Sec. 12.1.1 is an
example.) Thus, the time derivatives ∂
t
ξ and ∂
t
q are replaced by
D
t
q = ∂
t
q + Gq, D
t
ξ = ∂
t
ξ + Gξ. (12.21)
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288 Gauge theory of ideal ﬂuid ﬂows
Correspondingly, the Lagrangian L
f
of (12.13) is replaced by
L
f
=

Λ
f
(v)d
3
x :=
1
2

'v, v`d
3
a, (12.22)
v = D
t
x
a
, (12.23)
where d
3
a = ρd
3
x denotes the mass (in place of m
j
) in a volume
element d
3
x of the x-space with ρ the mass-density.
6
The action is
deﬁned by
I =

t
2
t
1
L
f
dt =

t
2
t
1
dt

M
d
3
xΛ
f
(ρ, D
t
x
a
), (12.24)
where M is a bounded space of E
3
, and Λ
f
=
1
2
ρ'D
t
x
a
, D
t
x
a
`.
12.4. Symmetry of ﬂow ﬁelds I: Translation symmetry
The symmetries of ﬂuid ﬂows we are going to investigate are the
translation symmetry and rotation symmetry, seen in the discrete
system.
The Lagrangian (12.13) has a global symmetry with respect to
the translational transformations (and possibly with respect to rota-
tional transformations). A family of translational transformations
is a group of transformations,
7
i.e. a translation group. Lagrangian
deﬁned by (12.22) for a continuous ﬁeld has the same properties
globally, inheriting from the discrete system of point masses. It is
a primary concern here to investigate whether the system of ﬂuid
ﬂows satisﬁes local invariance. First, we consider parallel transla-
tions (without local rotation), where the coordinate q
i
is regarded as
6
Here the Lagrangian coordinates a = (a, b, c) are deﬁned so as to represent the
mass coordinate. Using the Jacobian of the map x → a deﬁned by J = ∂(a)/∂(x),
we have d
3
a = Jd
3
x, where J is ρ.
7
A family of transformations is called a group G, provided that, (i) with two
elements g, h ∈ G, their product g · h is another element of G (in the case of
translations, g · h ≡ g +h), (ii) there is an identity e ∈ G such that g · e = e· g = g
for any g ∈ G (e = 0 in the case of translations), and (iii) for every g ∈ G, there is
an inverse element g
−1
∈ G such that g · g
−1
= g
−1
· g = e (g
−1
= −g in the case
of translations). If g · h = h· g for any g, h ∈ G, the group is called a commutative
group, or an Abelian group.
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12.4. Symmetry of ﬂow ﬁelds I: Translation symmetry 289
the cartesian space coordinate x
k
= x
k
a
(t) (kth component), and q
i
t
is a velocity component u
k
= ∂
t
x
k
a
(t).
12.4.1. Translational transformations
Suppose that we have a diﬀerentiable function f(x). Its variation by
an inﬁnitesimal translation x → x + ξ is given by δf = ξ∂
x
f where
∂
x
≡ ∂/∂x is regarded as a translation operator (ξ a parameter).
The operator of parallel translation in three-dimensional cartesian
space is denoted by T
k
= ∂/∂x
k
= ∂
k
, (k = 1, 2, 3). An arbitrary
translation is represented by ξ
k
T
k
(≡
¸
3
k=1
ξ
k
T
k
) with ξ
k
inﬁnitesi-
mal parameters. For example, a variation of x
i
is given by
δx
i
= (ξ
k
T
k
)x
i
= ξ
k
δ
i
k
= ξ
i
,
since T
k
x
i
= ∂
k
x
i
= δ
i
k
. If the Lagrangian density Λ of (12.15) is
independent of the coordinate q (such as Λ
f
of (12.22)), then Λ is
invariant with respect to the translational transformation, i.e. Λ has a
translational symmetry.
8
If ξ
k
are constants, the symmetry is global,
whereas if ξ
k
are functions of x and t, the symmetry is local. This
implies that the covariant derivative may be of the form:
d
t
= ∂
t
+ g
k
T
k
, (12.25)
for the translational symmetry, where g
k
(k = 1, 2, 3) are scalars.
This is investigated just below.
12.4.2. Galilean transformation (global)
Translational transformation from one frame F to another F

mov-
ing with a relative velocity U is called the Galilean transformation
in Newtonian mechanics. The transformation law is expressed as
follows:
x ≡ (t, x) ⇒x

). (12.29)
12.4.3. Local Galilean transformation
Suppose that a velocity ﬁeld u(x, t) is deﬁned by the velocity
(d/dt)x
a
(t) of ﬂuid particles, i.e. u(x
a
, t) = (d/dt)x
a
(t). Consider
the following inﬁnitesimal transformation:
x

(x, t) = x +ξ(x, t), t

= t. (12.30)
This is regarded as a local gauge transformation between noninertial
frames. In fact, the transformations (12.30) is understood to mean
that the coordinate x of a ﬂuid particle at x = x
a
(t) in the frame F
is transformed to the new coordinate x

of F

, which is given by x

=
x

a
(x
a
, t) = x
a
(t) + ξ(x
a
, t). Therefore, its velocity u = (d/dt)x
a
(t)
is transformed to the following representation:
u

k
− T
k
=
−(T
k
ξ
l
)T
l
), its transformation is given by
δ(d
t
q
i
) = δ(∂
t
q
i
) + (δu
k
)T
k
q
i
+ u
k
(δT
k
)q
i
+ u
k
T
k
(δq
i
)
= (dξ
k
/dt)T
k
q
i
−u
k
(T
k
ξ
l
)T
l
q
i
+ u
k
T
k
ξ
i
= (dξ
i
/dt), (12.41)
where T
k
q
i
= δ
i
k
is used. Thus, it is found that, in this gauge transfor-
mation, the covariant derivative d
t
q transforms just like the velocity
ﬁeld u whose transform is given by (12.38) (or (12.31)). This implies
that d
t
q represents in fact the particle velocity. Namely,
d
t
x = (∂
t
+ u
k
∂
k
)x = u(x, t) (12.42)
where u(x
a
, t) = (d/dt)x
a
(t) (see the beginning of Sec. 12.4.3).
12.4.5. Galilean invariant Lagrangian
When a representative ﬂow velocity v becomes higher to such a
degree that it is not negligible relative to the light velocity c, the
Galilean transformation must be replaced by the Lorentz transforma-
tion (Appendix F, or [LL75]). Theory of quantum ﬁelds is formulated
under the framework of the Lorentz transformation (F.1), whereas
ﬂuid mechanics is constructed under the above Galilean transfor-
mation (12.26). The diﬀerence is not an essential obstacle to the
formulation of gauge theory, because the Galilean transformation is
considered to be a limiting transformation of the Lorentz transfor-
mation of space-time as β = v/c →0.
In Appendix F.1, the Lagrangian Λ
G
of ﬂuid motion in the
Galilean system is derived from the Lorentz-invariant Lagrangian
Λ
(0)
L
by taking the limit as v/c → 0. Thus, the Lagrangian in the
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12.4. Symmetry of ﬂow ﬁelds I: Translation symmetry 293
Galilean system is given by the expression (F.6), which is reproduced
here:
Λ
G
dt = dt

M
d
3
xρ(x)

1
2
'u(x), u(x)` −(ρ, s)

, (12.43)
where ρ is the ﬂuid density, and M is a bounded space under con-
sideration with x ∈ M ⊂ R
3
. Namely, the Lagrangian L

deﬁned by
L

= −

M
(ρ, s)ρd
3
x, (12.44)
must be introduced for the invariance, where (ρ, s) is the internal
energy and s the entropy. It is understood that the background con-
tinuous material is characterized by the internal energy of the ﬂuid,
given by a function (ρ, s) where and s are deﬁned per unit mass.
10
For the ﬁelds of density ρ(x) and entropy s(x), the Lagrangian L

is invariant with respect to the gauge transformation (12.30), since
the transformation is a matter of the coordinate origin under the
invariance of mass: ρ(x)d
3
x = ρ

(x

)d
3
x

and the coordinate x does
not appear explicitly.
According to the scenario of the gauge principle, an additional
Lagrangian (called a kinetic term, [Fr97]) is to be deﬁned in con-
nection with the background ﬁeld (the material ﬁeld in motion in
the present context), in order to get nontrivial ﬁeld equations (for ρ
and s). Possible type of two Lagrangians are proposed as
L
φ
= −

M
d
t
φρd
3
x, L
ψ
= −

M
d
t
ψρsd
3
x (12.45)
where φ(x, t) and ψ(x, t) are scalar gauge potentials associated with
the density ρ and entropy s respectively.
11
These Lagrangians are
invariant by the same reasoning as that of L

above. In particular, d
t
is invariant by local Galilean transformation. It will be found later
10
In thermodynamics, a physical material of a single phase is characterized by
two thermodynamic variables such as ρ, s etc., which are regarded as gauge ﬁelds
in the present formulation.
11
The minus signs in L’s are a matter of convenience, as become clear later.
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294 Gauge theory of ideal ﬂuid ﬂows
that the equations of mass conservation and entropy conservation
are deduced as the results of variational principle. Thus the total
Lagrangian is deﬁned by L
T
:= L
f
+ L

i
= R
ij
v
j
in component representation,
where R = (R
ij
) is a matrix of rotational transformation. Rotational
transformation requires that the magnitude [v[ is invariant, that is
isometric. Therefore, we have the invariance of the inner product:
'v

, v

` = 'Rv, Rv` = 'v, v`. This is not more than the deﬁnition of
the orthogonal transformation. In matrix notation,
'v

, v

` = (v

)
i
(v

)
i
= R
ij
v
j
R
ik
v
k
= v
k
v
k
= 'v, v`.
Therefore, the orthogonal transformation is deﬁned by
R
ij
R
ik
= (R
T
)
ji
R
ik
= (R
T
R)
jk
= δ
jk
, (12.46)
where R
T
is the transpose of R: (R
T
)
ji
= R
ij
. Using the unit matrix
I = (δ
jk
), this is rewritten as
R
T
R = RR
T
= I. (12.47)
Hence, R
T
is equal to the inverse R
−1
of R.
There is a group of special signiﬁcance, which is SO(3).
12
When
we mention a rotational symmetry of ﬂuid ﬂows, we consider
12
SO(3): Special Orthogonal group. An element g ∈ SO(3) is characterized by
det(g) = 1. This is a subgroup of a larger orthogonal group satisfying (12.47),
which leads to det(g) = ±1. If the inner products such as A, A, B, B, etc. are
invariant, then invariance of the cross inner product A, B can be veriﬁed from
the invariance of A + B, A + B = A, A + 2A, B +B, B.
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12.5. Symmetry of ﬂow ﬁelds II: Rotation symmetry 295
SO(3) always because it is connected with the identity transforma-
tion R = I.
It is almost obvious now that the Lagrangian density Λ
f
of (12.22)
is invariant under the rotational transformation of SO(3), v(x) →
v

M
dt d
3
x Λ
T
= 0. (12.75)
Usually, in the variational formulation (of the Eulerian representation
of independent variables x and t), the Euler’s equation of motion is
derived under the constraints of the continuity equation and isen-
tropic equation. In the present analysis, the variational principle
based on the gauge principle gives us the continuity equation and
the isentropic equation as outcomes of variations of the Lagrangian
L
T
[v, ρ, s, φ, ψ, b] with respect to variations of the gauge potentials
φ and ψ. The potentials have intrinsic physical signiﬁcance in the
framework of the gauge theory.
Here, we have to take into consideration a certain thermodynamic
property that the ﬂuid is an ideal ﬂuid in which there is no mechanism
of dissipation of kinetic energy into heat. That is, there is no heat
production within ﬂuid. By thermodynamics, change of the internal
energy and enthalpy h = + p/ρ can be expressed in terms of
changes of density δρ and entropy δs as
δ =

= ∂
t
v + (v ∇)v = D
t
v. (12.95)
It is interesting to ﬁnd that the ﬂuid motion is driven by the velocity
potential Φ, where Φ = φ + s
0
ψ with φ, ψ the gauge potentials.
We recall that the ﬂow of a superﬂuid in the degenerate ground
state, which is characterized with zero entropy, is represented by
a velocity potential in Chapter 11. There, it is shown that such a
ﬂuid of macroscopic number of bosons is represented by a single
wave function in the degenerate ground state, where the quantum-
mechanical current is described by a potential function (phase of the
wave function). Therefore the corresponding velocity is irrotational.
In this case, local rotation would not be captured.
12.6.6. Clebsch solution
If the entropy s is not uniform (i.e. a function of points and time)
but in case that the vector ﬁeld b is still absent, the present solution
is equivalent to the classical Clebsch solution owing to the property
D
t
ψ = 0. From (12.89) and (12.90), the velocity and vorticity are
v = ∇φ + s∇ψ, ω = ∇s ∇ψ. (12.96)
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306 Gauge theory of ideal ﬂuid ﬂows
In this case, the vorticity is connected with nonuniformity of entropy.
In addition, Eq. (12.84) can be written as
1
2
v
2
+ h + ∂
t
φ + s∂
t
ψ = 0, (12.97)
because we have
D
t
φ + sD
t
ψ = ∂
t
φ + s∂
t
ψ +v (∇φ + s∇ψ) = ∂
t
φ + s∂
t
ψ +v
2
.
It is shown in Problem 12.1 that Euler’s equation of motion,
∂
t
v +ω v = −∇

1
2
v
2
+ h

, (12.98)
is satisﬁed by Eq. (12.97) owing to the conditions (12.85) and
(12.88), under the deﬁnition (12.96) and the barotropic relation
h(p) =

p
dp

/ρ(p

). In this case, the helicity vanishes (Problem 12.1).
Equation (12.98) can be written also as
∂
t
v + (v ∇)v = −
1
ρ
grad p, (12.99)
because of the identity: ω v = (v ∇)v −∇(
1
2
v
2
) and the relation
(12.77) under (12.88).
12.7. Variations and Noether’s theorem
Diﬀerential form of momentum conservation results from the Noether
theorem associated with local translational symmetry. Here, varia-
tions are taken with respect to translational transformations only,
with the gauge potentials ﬁxed. As before, the action I is deﬁned by
I =

ξ
k
,
(12.118)
where S is the boundary surface of M and (n
l
) = n is a unit outward
normal to S. The terms on the second line are integrated terms,
which came from the second line of (12.117). These vanish owing to
the imposed conditions (12.107) and (12.108).
Thus, the invariance of I for arbitrary variation of ξ
k
satisfying
the conditions (12.107) and (12.108) results in
∂
t
(ρv
k
) + ∂
l

, (12.122)
for k = 1, 2, 3. This is the conservation of total momentum, equivalent
to (3.23) without the external force f
i
.
12.8. Additional notes
12.8.1. Potential parts
From the equation of motion (12.120), the vorticity equation (7.1)
can be derived by taking its curl:
∂
t
ω + curl(ω v) = 0, (7.1)
for a homentropic ﬂuid (of uniform entropy). Suppose that this is
solved and a velocity ﬁeld v(x, t) is found. Then, the velocity v should
satisfy the following equation:
∂
t
v +ω v = ∇H
∗
for a scalar function H
∗
(x, t). Obviously, taking curl of this equation
reduces to (7.1). On the other hand, the equation of motion (12.120)
was transformed to (3.30) for a homentropic ﬂuid in Sec. 3.4.1:
∂
t
v +ω v = −grad

M
ρ'b, v`d
3
x. In Appendix F.2, the same Lagrangian is repre-
sented as
L
A
= −

M
'L
X
A
1
, ω`d
3
x = −

M
'∇(L
X
A
1
), v` d
3
x. (12.125)
where A
1
is a gauge potential 1-form (deﬁned there), and X is a tan-
gent vector deﬁned by ∂
t
+v
k
∂
k
, which is equivalent to the operator
of convective derivative D
t
as far as its form is concerned.
16
Thus,
the vector ﬁeld b is derived from a vector gauge potential A
1
by the
relation ρb = ∇(L
X
A
1
), where L
X
A
1
denotes the Lie derivative of
16
For a scalar function f (0-form), the Lie derivative of f is given by L
X
f =
Xf = D
t
f. However, if applied to other forms or vectors F (say), L
X
F is diﬀerent
from D
t
F = ∂
t
F + v
k
∂
k
F, where D
t
is a simple diﬀerential operator D
t
. In this
regard, concerning the terms D
t
φ and D
t
ψ of (12.74), it is consistent and more
appropriate to write L
X
φ and L
X
ψ, which are equivalent to D
t
φ and D
t
ψ since
φ and ψ are scalar functions.
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12.9. Problem 313
the form A
1
along the ﬂow generated by X. In the Appendix, diﬀer-
ential calculus are carried out in order to elucidate the signiﬁcance
of the gauge potential A
1
associated with the rotation symmetry.
It is remarkable that the action principle for arbitrary variations
of A
1
results in the vorticity equation (Appendix F.2):
∂
t
ω + curl(ω v) = 0 .
Thus, it is found that the vorticity equation is an equation of a gauge
ﬁeld.
12.9. Problem
Problem 12.1 Clebsch solution
Verify that the Euler’s equation of motion for a barotropic ﬂuid of
p = p(ρ),
∂
t
v +ω v = −∇

. (A.17)
As is evident from the deﬁnition of the determinant, we have the
equality, A· (B×C) = B· (C×A) = C· (A×B). This is equal to
the volume of a hexahedron composed of three vectors A, B, C.
A vector triple product is deﬁned by
[A×(B×C)]
i
= A
k
B
i
C
k
−A
k
B
k
C
i
, (A.18)
which is an ith component, and summation with respect to k is
understood. If A, B, C are ordinary vectors, then rearranging the
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318 Vector analysis
order, we have
A×(B×C) = (A· C)B−(A· B)C. (A.19)
A particularly useful expression is obtained when we set
A = C = v and B = ∇ = (∂
1
, ∂
2
, ∂
3
) in (A.18), which is
[v ×(∇×v)]
i
= v
k
∂
i
v
k
−v
k
∂
k
v
i
= ∂
i

C
B· dl, (A.35)
where A is an open surface bounded by a closed curve C of a line-
element dl, and n is a unit normal to the surface element dA.
(iii) Gauss’s theorem for a vector ﬁeld A(x):

V
div AdV =

S
A· n dS, (A.36)
where S is a closed surface bounding a volume V , and n is a unit
outward normal to S.
A.7. δ function
The delta function δ(x) is deﬁned as follows. For an interval D of
a variable x including x = 0, suppose that the following integral
formula is valid always,

, (B.8)
where C is an anti-clockwise closed curve bounding a two-
dimensional domain D.
When (A
x
, A
y
) is replaced by (u, v), the left-hand side vanishes
owing to (B.6). Hence, we have
0 =

C

−v dx + udy

=

C
w· dl, w := (−v, u), (B.9)
where a new vector ﬁeld w is deﬁned.
Obviously, there is analogy between the above (B.9) and Eq. (B.2)
in the previous section. By the same reasoning as before, it can be
shown that there exists a function Ψ(x, y) deﬁned by
Ψ(x, y) =

dt. (F.4)
The third −c
2
dt term is necessary so as to satisfy the Lorentz-
invariance ([LL75], Sec. 87). The reason is as follows. Obviously, the
term v(x), v(x) is not invariant with the Galilei transformation,
v → v

= v − U. Using the relations dx = vdt and dx

= v

dt =
(v −U)dt

, the invariance (F.2) leads to
c
2
dt

= c
2
dt +

−v, U +
1
2
U
2

1 + O((v/c)
2
)

dt (F.5)
The second term on the right makes the Lagrangian Λ
(0)
L
dt Lorentz-
invariant exactly in the O((v/c)
0
) terms in the limit as v/c → 0.
When we consider a ﬂuid ﬂow as a Galilean system, the following
prescription is applied. Suppose that the ﬂow is investigated in a
ﬁnite domain M in space. Then the c
2
dt term gives a constant c
2
Mdt
to Λ
(0)
L
dt, where M =

M
d
3
xρ(x) is the total mass in the domain
M. In carrying out variation of the action I, the total mass M is
ﬁxed at a constant.
Now, keeping this in mind implicitly, we deﬁne the Lagrangian
Λ
G
of a ﬂuid motion in the Galileian system by
Λ
G
dt =

V
ρ
−1
(ω · L
X
B)d
3
x.
Thus it is found that the term L
X
B generates the helicity.
Density of the Lagrangian L
A
is deﬁned by Λ
A
:= −Ψ, ω:
Λ
A
:= −Ψ, ω = −ω
i
∂
t
A
i
−ω
i
v
k
∂
k
A
i
−ω
i
A
k
∂
i
v
k
.
Rearranging the ﬁrst two terms, we obtain
Λ
A
=

∂
t
ω
i
+ ∂
k
(v
k
ω
i
) −ω
k
∂
k
v
i

A
i
−∂
t
(ω
i
A
i
) −∂
k
(ω
i
v
k
A
i
)
Next, we consider the action I
A
=

t
2
t
1
dt

M
Λ
A
d
3
x and its variation
with respect to the variation δA. Applying the same boundary con-
ditions (12.107) and (12.108) to δA
i
(used for the variations of ξ),
we obtain
δI
A
=

t
2
t
1
dt

M

∂
t
ω
i
+ ∂
k
(v
k
ω
i
) −ω
k
∂
k
v
i
] δA
i
d
3
a.
Invariance of I
A
for arbitrary variations δA
i
implies
∂
t
ω
i
+ v
k
∂
k
ω
i
+ ω
i
∂
k
v
k
−ω
k
∂
k
v
i
= 0.
Thus, we have found the equation for the gauge ﬁeld ω. In vector
notation, this is written as
∂
t
ω + (v · ∇)ω + ω(∇· v) −(ω · ∇)v = 0. (F.15)
This is nothing but the vorticity equation, which is also written as
∂
t
ω + curl(ω ×v) = 0 . (F.16)
November 1, 2006 2:8 WSPC/Book -SPI-B364 “Elementary Fluid Mechanics” Trim Size for 9in x 6in solutions
Solutions
Problem 1
1.1 : This is not the particle path, since the pattern is a snap-shot. It is not
the stream-line either because there is no physical reason (no physical
constraint) that the ﬂuid is moving in the tangent direction of the
line. The ink-particles composing the lines are not the family that
passed at a ﬁxed point at successive times. Therefore the pattern is
not the streak-line.
The pattern is interpreted as a diﬀeomorphism mapping from the
compact domain occupied by the initial ink distribution on the surface
onto the present domain of streaky pattern which is formed by the
straining surface deformation (stretching in one direction and com-
pression in the perpendicular direction) and folding of the elongated
domain. Thus initial drop-like domain is mapped in diﬀeomorphic
way to a streaky domain by the motion of the water surface with
eddies.
1.2 : Consider a ﬂuid particle located at x at a time t, and denote its
subsequent position at a time t+δt by T(x) which is given by x+vδt+
O(δt
2
). Suppose that the line element δx between two neighboring
points at x and x + δx is represented by δX at a subsequent time
t + δt, where δx = (δx, δy, δz)
T
with
T
denoting the transpose. The
new line element δX = (δX, δY, δZ)
T
will be given by
δX = T(x +δx) −T(x)
= (x +δx −x) + (v(x +δx) −v(x))δt +O(δt
2
)
= δx +

∞
−∞
u
0
(ξ) δ(x − ξ)dξ = u
0
(x),
satisfying the initial condition.
2.4 : Taking the coordinate origin at the center of the cube of side a, we
consider the equation of the angular momentum of the cube. Denoting
the moment of inertia of the cube around the z axis by I
z
, and the
angular velocity around it by Ω
z
which is assumed uniform over the
cube, the z component angular momentum is given by M
z
= I
z
Ω
z
,
where I
z
=

(x
2
+ y
2
)ρ dxdydz. The dynamical equation is given
by (d/dt)(I
z
Ω
z
) = N
z
, where N
z
is the moment of the stress force
over the faces from the outside ﬂuid. To the ﬁrst approximation,
the total surface forces exerted on the faces perpendicular to the
x axis are f
(+)
x
=(σ
xx
, σ
yx
, σ
zx
) a
2
for the face of positive x, and
f
(−)
x
= −(σ
xx
, σ
yx
, σ
zx
) a
2
for the face of negative x. Hence, the z
component of the moment of the stress forces is given by N
(x)
z
=
1
2
a

.
Its real part yields the solution (4.79). The representative thickness
δ of the oscillating boundary layer will be given by the y-value cor-
responding to the place where the amplitude decays to 1/e of the
boundary value. Thus, δ =

∞
−∞
u
2
dy is invariant
with respect to x. This integral multiplied with ρ denotes a physi-
cally signiﬁcant quantity, i.e. the total momentum ﬂux M across a
ﬁxed x position. Invariance of M means that there is no external
momentum source in the x direction under uniform p.
(iii) Equation (4.86) is written as f

, (A)
where Q is a constant whose meaning will become clear just below.
Carrying out the integral of u(x, y) with respect to y, we obtain

∞
−∞
u(x, y)dy
= Q

U
4πνx

∞
−∞
exp

−
Uy
2
4νx

dy = Q,
where the integration formula

∞
−∞
exp(−η
2
) dη =
√
π is used.
Thus, it is found that the mass ﬂux defect at a ﬁxed x position is
a constant Q independent of x.
(ii) We shall use the momentum equation in the integral form (3.23)
where P
ik
= ρv
i
v
k
+ pδ
ik
. For the control surface A
0
, we choose
a large rectangular box with sides parallel and perpendicular to
the undisturbed uniform stream, including the body inside. The
sides S of A
0
parallel to the stream are located far enough from
the body to lie entirely outside the wake. The ﬂuid within A
0
is
acted on by forces on the surrounding surface A
0
, and by forces
at the body surface whose resultant should be −D (sink) in the
x direction.
Under the steady ﬂow assumption, the drag D is given by
D =

F

p
1
+ρu
2
1
−p
2
−ρu
2
2

dF −ρ

S
uv
n
dS, (B)
where u
1
, p
1
and u
2
, p
2
are the x-velocity and the pressure at
the upstream and downstream faces F, respectively. The second
integral term is the x-momentum ﬂux out of the side surface S (so
that A
0
= F + S), where the pressure eﬀect cancels out on both
sides. All the viscous stresses on A
0
are assumed to be small, and
so neglected.
November 1, 2006 2:8 WSPC/Book -SPI-B364 “Elementary Fluid Mechanics” Trim Size for 9in x 6in solutions
Problem 4 351
The mass ﬂux across A
0
must be exactly zero, which is given by

F
(u
2
−u
1
)dF +

S
v
n
dS = 0. (C)
We may put u = U in the second term of (B). With the aid of
(C), we have
D =

F
(p
1
+ρu
1
(u
1
− U) −p
2
−ρu
2
(u
2
−U))dF. (D)
At the far downstream face, the diﬀerence from the far upstream
lies in the existence of the wake, regarded as an “in-ﬂow” towards
the body. The volume ﬂux of this “in-ﬂow” is equal to Q calcu-
lated in (i). This in-ﬂow in the wake must be compensated by an
equal volume ﬂux away from the body outside the wake where
the ﬂow is practically irrotational. Thus, the presence of the wake
is associated with a source-like ﬂow (Fig. 4.12), with its strength
being Q. At large distance r from the body, the source-ﬂow veloc-
ity v falls oﬀ as r
−1
in two-dimensions (see (5.63) of Sec. 5.8.1).
In this same region, the Bernouli’s theorem (5.12) of Sec. 5.1 is
applicable so that we have p
1
−p
2
= −
1
2
ρ((v
1
)
2
−(v
2
)
2
).
Thus, far from the body, the ﬂow is a superposition of a uniform
stream and the source-like ﬂow outside the wake, and within the
wake, we have the velocity (A) obtained in (i). In this situation,
most terms in the expression (D) cancel, and the only remaining
contribution is obtained from the wake, given by D = ρU

sinφ. (A)
(i) Setting σ = a, we obtain x = 2a cos φ, and y = 0. For φ ∈ [0, 2π],
we have x ∈ [−2a, 2a].
The exterior of the circle σ = a is expressed by σ > a and
φ ∈ [0, 2π], which corresponds to the whole z-plane with the cut
L. According to (A), counter-clockwise rotation around ζ = 0
corresponds to the same counter-clockwise rotation around L.
The interior of the circle σ = a corresponds to the whole z
plane with the cut L, again. But this time, the counter-clockwise
rotation around ζ = 0 corresponds to the clockwise rotation
around L, according to (A).
(ii) As ζ → ∞, F
α
→ Ue
−iα
ζ. Hence, the ﬂow tends to an inclined
uniform ﬂow with an angle α counter-clockwise with the real axis
of the ζ-plane. The second term, which was neglected at inﬁnity,
is a dipole Ua
2
e
iα
/ζ with its axis inclined by an angle π+α with
the positive real axis of ξ. Substituting ζ = σe
iφ
into (5.89), we
November 1, 2006 2:8 WSPC/Book -SPI-B364 “Elementary Fluid Mechanics” Trim Size for 9in x 6in solutions
354 Solutions
obtain
F
α
=

σ +
a
2
σ

cos(φ −α) +i

σ −
a
2
σ

sin(φ −α).
This means that the circle σ = a is a stream-line. Thus, F
α
represents a uniform ﬂow around a circle of radius a, and the
ﬂow is inclined at an angle α with respect to the positive real
axis of ζ.
(iii) By the transformation (5.88), the circle σ = a collapses to the
double segments of L representing a ﬂat plate of length 4a. As
ζ → ∞, z ≈ ζ. Therefore, the ﬂow at inﬁnity of the z-plane is
also an inclined uniform ﬂow with an angle α counter-clockwise
with the real axis of z.
(iv) If α = π/2, the potential F
α
(ζ) reduces to the potential F
⊥
(ζ)
of (5.90). Hence, F
⊥
(ζ) represents a vertical uniform ﬂow from
below (upward) in the ζ-plane. Since z ≈ ζ as ζ → ∞, the
ﬂow in the z-plane is the same vertical uniform ﬂow from below
impinging on the horizontal ﬂat plate perpendicularly. Because
Z = −iz = ze
−iπ/2
, the Z-plane is obtained by rotating the
z-plane by 90
◦
clockwise. Thus, F
⊥
represents a ﬂow impinging
on a vertical ﬂat plate at right angles from left in the Z-plane.
5.6 : If n = −1,

ω(y)
[x −y[
since ∇
y
ω(y) = 0, and ∇
x
= −∇
y
if operated to 1/[x − y[. The
last integral is transformed to a surface integral which vanishes by
(7.4a).
7.2 : (a) We take time derivative of the impulse P of (7.9) and use the
equation (7.1). Then we obtain
d
dt
P =
1
2

D
x ∂
t
ωd
3
x =
1
2

D
x (∇q) d
3
x
where q = −ωv. Using (7.5e) with v
i
replaced by q
i
, the right-
hand side reduces to

, can
be written as
(x q)
i
= ε
ijk
x
j
∂
l
V
lk
= ε
ijk
∂
l
(x
j
V
lk
),
since ε
ijk
V
jk
= 0. Thus, the integral of (B) can be converted
to vanishing surface integrals, and we obtain invariance of L:
(d/dt)L = 0.
(c) Regarding the helicity H, let us consider the following identity
for two arbitrary vectors A and B: ∇ (AB) = (∇A) B−
A (∇ B). When A and B decay at inﬁnity like the vector v,
we have the equality:

l = Ul.
For the vorticity ω = (0, 0, ω ) integral is taken over the area of
S = 2εl of the rectangle abcd:
Γ[C] =

l
0
dx

ε
−ε
dy ω
z
(x, y) = Ul.
Since this holds for any l and ε, this implies that

ε
−ε
dyω
z
(x, y) = U
must hold for any ε. This leads to ω
z
= U δ(y). If the contour C
does not cut the x axis, we have Γ[C] = 0. This is also satisﬁed by
ω
z
= Uδ(y).
7.5 : Taking s = 0 at the origin O, i.e. y(0) = 0, the position y(s) on the
curve T near O is expanded in the frame K as
y(s) = y

[ρ
1
−ρ
2
[
ρ
1
+ρ
2
gk (> 0).
(ρ
1
> ρ
2
). Hence, σ = ±σ
∗
. Therefore, the solution includes a
growing mode σ = +σ
∗
, and the basic state is unstable. (This is
called the Rayleigh–Taylor instability.)
(iii) Even if a light ﬂuid is placed above a heavy ﬂuid (ρ
1
< ρ
2
), we
have the same perturbation equations, but we obtain σ
2
= −σ
2
∗
,
and we have σ = ±iσ
∗
. Therefore, the separation surface oscillates
with the angular frequency σ
∗
.
[This is called the interfacial wave, observed in the ocean (or at
the estuary) at the interface of salinity discontinuity. Often this is
also seen in a heated room at temperature discontinuity visualized
by the tabacco smoke.]
9.2 : (i) Multiplying both sides of Eq. (9.26) by the complex conjugate φ
∗
of φ, and integrating from −b to b, and performing integration by
parts for the ﬁrst term, we obtain

b
−b
([φ

[
2
+k
2
[φ[
2
)dy +

b
−b
U

U −c
[φ[
2
dy = 0,
by using the boundary condition (9.24). Setting c = c
r
+ ic
i
and
taking the imaginary part of the above expression, we obtain
c
i

b
−b
U

(y)
[U −c[
2
[φ[
2
dy = 0,
since the ﬁrst term is real. For the instability, we should have
c
i
> 0, which means c
i
= 0 anyway. Therefore, the integral
must vanish. For its vanishing, U

(y) must change its sign. Thus,
the proﬁle U(y) should have inﬂexion-points (where U

(z)w.
In order to obtain (9.76), we apply ∇
2
to the third equation
(9.73) and eliminate R
e
∇
2
p by using the above equation. Thus,
November 1, 2006 2:8 WSPC/Book -SPI-B364 “Elementary Fluid Mechanics” Trim Size for 9in x 6in solutions
Problem 10 367
we obtain ﬁnally (9.76) for w only. The boundary condition is the
usual no-slip condition at the walls.
(ii) Noting that the Reynolds number R
e
is included always as a com-
bination αR
e
and the y wave number β is included in ∇
2
as a com-
bination

α
2
+β
2
in (9.76), the eigenvalue problem determines
the exponential growth rate as a function of αR
e
and

α
2
+β
2
), which should be positive. The
critical Reynolds number R
c
is by deﬁnition the lowest value of
such R
e
for all possible values of the wavenumbers α and β, thus
represented by (9.78).
(iii) Critical Reynolds number R
2D
c
for two-dimensional problem is
represented by (9.78) with β = 0:
R
2D
c
= min
α
(F(α)/α).
The critical Reynolds number considered in the previous (ii) is
R
3D
c
. Using k =

∞
0
k
2
E(k)dk,
since
1
2
[ˆ u(k)[
2
= Φ([k[) by (10.12), and Φ([k[) 4π[k[
2
= E([k[)
by (10.13) for isotropic turbulence.
10.2 : (i) Choosing two arbitrary constant vectors (a
i
) and (b
j
), we take
scalar product (i.e. contraction) with B
ij
(s). The resulting
B
ij
a
i
b
j
is a scalar and possibly depends on six scalar quantities
(only): (s
i
s
i
), (a
i
s
i
), (b
i
s
i
), (a
i
a
i
), (b
i
b
i
) and (a
i
b
i
) by trans-
formation invariance of scalars. In addition, B
ij
a
i
b
j
should be
bilinear with respect to a
i
and b
j
, such as (a
i
b
i
) or (a
i
s
i
)(b
j
s
j
).
Furthermore, B
ij
a
i
b
j
may depend on scalar functions of the
form F
1
(s) or G(s). The B
ij
a
i
b
j
satisfying these can be rep-
resented as
B
ij
(a
i
b
j
) = G(s)(a
i
b
i
) +F
1
(s)(a
i
s
i
)(b
j
s
j
)
= (G(s)δ
ij
+F
1
(s)s
i
s
j
)a
i
b
j
.
This implies (10.40) with F(s) = F
1
(s)s
2
.
(ii) Using ∂u
j
(x

+G(k)
k sin ks
s
Integrating H(k) from k = 0 to ∞, the ﬁrst term vanishes since
G(∞) = 0, and the second gives (10.52).
(iii) The left-hand side of (10.53) can be written as (e
iks
= cos ks +
i sinks),
1
2π

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This book aims to provide an elementary interpretation on physical aspects of ﬂuid ﬂows for beginners of ﬂuid mechanics in physics, mathematics and engineering from the point of view of modern physics. Original manuscripts were prepared as lecture notes for intensive courses on Fluid Mechancis given to both undergraduate and postgraduate students of theoretical physics in 2003 and 2004 at the Nankai Institute of Mathematics (Nankai University, Tianjin) in China. Beginning with introductory chapters of fundamental concepts of the nature of ﬂows and properties of ﬂuids, the text describes basic conservation equations of mass, momentum and energy in Chapter 3. The motions of viscous ﬂuids and those of inviscid ﬂuids are ﬁrst considered in Chapters 4 and 5. Emphasizing the dynamical aspects of ﬂuid motions rather than static aspects, the text describes, in subsequent chapters, various important behaviors of ﬂuids such as waves, vortex motions, geophysical ﬂows, instability and chaos, and turbulence. In addition to those fundamental and basic chapters, this text incorporates a new chapter on superﬂuid and quantized vortices because it is an exciting new area of physics, and another chapter on gauge theory of ﬂuid ﬂows since it includes a new fundamental formulation of ﬂuid ﬂows on the basis of the gauge theory of theoretical physics. The materials in this book are taken from the lecture notes of intensive courses, so that each chapter in the second half may be read separately, or handled chapter by chapter.

v

vi

Preface

This book is written with the view that ﬂuid mechanics is a branch of theoretical physics. June 2006 Tsutomu Kambe Former Professor (Physics), University of Tokyo Visiting Professor, Nankai Institute of Mathematics (Tianjin, China)

it is understood that plant
1
. a number of particles compose the body of matter. and are evolving with time always. On the other hand. jetliners and rockets utilize ﬂows in order to obtain thrust to move from one place to other while carrying loads. solar wind. ﬂows of microscopic suspension particles in a chemical test-tube. swimming ﬁshes are considered to be using water motions (eddies) to get thrust for their motion. From the technological aspect. and are moving and continuously changing their relative positions. and so on. it implies usually time development of the displacement and deformation of matter. Giving typical examples. vehicles such as ships. Animals such as insects or birds commonly use air ﬂows in order to get lift for being airborne as well as getting thrust for their forward motion.
What are ﬂows ?
Fluid ﬂows are commonly observed phenomena in this world. ﬂows of blood or respiratory air in a body. gas ﬂows in interstellar space. aeroplanes.Chapter 1
Flows
1. Flows are observed in diverse phenomena in addition to the wind and river above: air ﬂows in a living room. ﬂows of bathtub water. from the biological aspect. When we say ﬂow of a matter. Namely. Glider planes or soaring birds use winds passively to get lift.1. In addition. the wind is a ﬂow of the air and the river stream is a ﬂow of water. the motion of clouds or smoke particles ﬂoating in the air can be regarded as visualizing the ﬂow that carries them. atmospheric ﬂows and sea currents. On the other hand.

A ﬂuid particle is deﬁned as a mass in a small nearly-spherical volume ∆V . whose diameter is suﬃciently small from a macroscopic point of view. a material which constantly deforms itself such as the air or water is called a ﬂuid. motions of material bodies of continuous mass distribution. 1. ﬂow of a ﬂuid. Every living organism has a certain internal system of physiological circulation. By contrast. however its deformation stops in balance with a force acting on it.e. Once the body is free from the force. A body of ﬂuid is composed of innumerably many microscopic molecules. it is large enough if it is compared with the intermolecular distance. However. the ﬂuid keeps deforming even when it is free from force. whereas the intermolecular scale is 10−6 mm or less.e. it might not be an exaggeration to say that all the living organisms make use of ﬂows in various ways in order to live in this world. such that the total number N∆ of molecules in the volume ∆V is suﬃciently big so that the statistical description makes sense. An elastic solid is deformable as well. often use wind for their purposes. i. A plastic solid is deformed continuously during the application of a force. In fact. At the normal temperature and pressure
. In general. i. Fluid particle and ﬁelds
When we consider a ﬂuid ﬂow. Fluid mechanics studies such ﬂows of ﬂuids. In general. the study of ﬂuid mechanics is normally carried out at scales of about 10−3 mm or larger. it is a basic assumption that there exists such a volume ∆V enabling to deﬁne the concept of a ﬂuid particle. under fundamental laws of mechanics. it stops deformation (nominally at least). However in a macroscopic world. or pollen. it is regarded as a body in which mass is continuously distributed. Once the body is released from the force. is considered to be a mass ﬂow involving its continuous deformation. it recovers its original state.2. In other words.2
Flows
seeds. Motion of a ﬂuid. it is often useful to use a discrete concept although the ﬂuid itself is assumed to be a continuous body.

y. where k is the Boltzmann constant.7 × 1016 molecules. and we consider the ﬂow in the (x.2. The ˜ diﬀerence uα = uα − v is called the peculiar velocity or thermal velocity. Position x = (x. Density of the ﬂuid ρ is deﬁned by dividing ∆M by ∆V . The ﬂuid α α α velocity v at x is deﬁned by the average value of the molecular velocities. Boltzmann constant k is a conversion factor between degree (Kelvin temperature) and erg (energy unit). Fluid particle and ﬁelds
3
(0◦ C and 760 mmHg). ∆V (1.1.
. α mα = mN∆ = ρ∆V .
(1. hence the mass in a small volume ∆V is ∆M = mN∆ . We consider a monoatomic gas whose molecular mass is m. say ux . deﬁned by k = 1. z) of a ﬂuid particle is deﬁned by the center of mass of the constituent molecules. the temperature T is deﬁned by the law that the average of peculiar kinetic energy per degree-offreedom is equal to 1 kT .2)
where mα = m (by the assumption).2 Each 2 molecule has three degrees of freedom for translational motion.1)
A α-th molecule constituting the mass moves with its own velocity uα . so that1 ρ(x) := ∆M . uy . uz . a cube of 1 mm in a gas contains about 2. where uα has three components. y. 2
A := B denotes that A is deﬁned by B. In the kinetic theory of molecules. It is assumed that 1 m(˜x )2 uα 2
1 2
=
1 m(˜y )2 uα 2
=
1 m(˜z )2 uα 2
1 = kT. and · denotes an average with respect to the molecules concerned. in such a way v(x) = uα :=
α mα uα α mα
.38 × 10−16 erg/deg. z) cartesian coordinate frame.

and n(˜x ) ˜β denotes the number of molecules with ux in a unit volume. The ﬁrst term of (1. the x-component of the pressure force Fx on ∆S acting from the left (smaller x) side to the right (larger x) ˜β side would be given by the ﬂux of x-component momentum mβ ux through ∆S: Fx = p(x) ∆S =
β
(mβ ux ) ux ∆S = ∆S ˜β ˜β
m(˜x )2 n(˜x ). u
where N is the total number of molecules in a unit volume. u u
ux ˜
(1. u u
(1. uy and uy +d˜y and uz and uz +d˜z ˜ ˜ u ˜ ˜ u ˜ ˜ u respectively. uz ). the factor ux m(˜ ) n(˜ ) on the right-hand side ˜ ˜ is expressed by the following two integrals for ux > 0 and ux < 0 ˜ respectively:
ux >0 ˜
˜ m(˜x )2 Nf (˜ ) d3 u + u u
ux <0 ˜
˜ m(˜x )2 Nf (˜ ) d3 u. This is interpreted as follows. uy . In the ˜ ux 2 ux kinetic theory. u with
˜ all u
˜ f (˜ ) d3 u = 1.5)
where the function f (˜ ) denotes the distribution function of the pecuu ˜ ˜ ˜ liar velocity u. uz ) d˜x d˜y d˜z .4
Flows
Therefore. is deﬁned by Nf (˜x . which are contained in the volume element ux ∆S. a force) through ∆S. The pressure p exerted on a surface element ∆S is deﬁned by the momentum ﬂux (i. uy .3)
α
On the other hand. uα 2 (1. u ˜ ˜ u u u
3
.4)
where β denotes all the molecules passing through ∆S per unit u time. u ˜ ˜ which takes values between ux and ux +d˜x . pressure is a variable deﬁned against a surface element. we have 3 kT (x) := 2 1 m˜ 2 uα 2 = 1 N∆ 1 m˜ 2 . while u u the second term denotes that a negative momentum m˜x (˜x < 0) is u u
More precisely. the number of molecules of the peculiar velocity (˜x .e.5) denotes that a positive momentum m˜x (˜x > 0) is absorbed into the right side of ∆S. and the number of molecules between u and u + d˜ u is given by3 ˜ Nf (˜ ) d3 u. Choosing the x-axis normal to the surface ∆S.

we obtain p(x) = NkT .3).4 The density ρ(x). t). where ρ = mN . Thus.314 × 107 erg/deg. a particle at x ∈ B moves from x to x + δx: x → x + δx = x + vδt + O(δt2 ) (1. After an inﬁnitesimal time δt. Both means that the space on the right side has received the same amount of positive momentum. z. and m 1 (˜ )2 u 3 3 u is given by kT from (1. z) and the time t smoothly. (1. temperature T (x) and pressure p(x) thus deﬁned depend on the position x = (x.
4
. ﬂow of a ﬂuid is regarded as a continuous sequence of mappings. y. Such variables are called ﬁelds.7)
This is known as the equation of state of an ideal gas. the initial domain B0 is mapped one after another smoothly and constantly.1.2. µm = mNA and R = NA k.4) and (1. and so on. From a mathematical aspect. Both terms are combined into one: mβ (˜x )2 = uβ
β ˜ all u
˜ m(˜x )2 Nf (˜ ) d3 u u u (1. Fluid particle and ﬁelds
5
taken out from the right side of ∆S. t). from (1. since the molecular kinetic motion usually works to smooth out discontinuity (if any) by the transport phenomena considered in Chapter 2. Subsequent mapping occurs for another δt from Bδt to B2δt . This point of view is often called the continuum hypothesis. y. u
u u u since 1 m (˜x )2 = 1 kT . At a later time t. Then the domain B0 may be mapped to Bδt (say).7) reduces to p = (1/µm )ρRT . Consider all the ﬂuid particles composing a subdomain B0 at an initial instant t = 0. In this way.023 × 1023 (Avogadro’s constant). these variables are regarded as continuous and in addition diﬀerentiable functions of (x.6). the domain
For a gram-molecule of an ideal gas.8)
by the ﬂow ﬁeld v(x.6)
= Nm (˜x )2 = NkT . Namely. For an ideal gas of molecular weight µm . N is replaced by NA = 6. the equation (1. velocity v(x). where the average (˜x )2 = (˜y )2 = 2 2 u (˜z )2 is equal to 1 (˜ )2 by an isotropy assumption. The product NA K = R is called the gas constant: R = 8.

Provided that the curve is represented as (x(s).9)
This system of ordinary diﬀerential equations can be integrated for a given initial condition at s = 0. t) − v(y. z(s)) in terms of a parameter s. t). z) w(x. z)) by the above deﬁnition. there is an inverse map. which should be parallel to the given vector ﬁeld (u(x.3. and that. w) is given in a subdomain of three-dimensional Euclidean space R3 . y(s). y. y. z) in the domain. w) is continuous and smooth at every point (x. y. a curve through the point P = (x(0). y. It is known in the theory of ordinary diﬀerential equations in mathematics that one can draw curves so that the curves are tangent to the vectors at all points. one of the simplest conditions is the Lipschitz condition: |v(x. t)| K|x − y| for a positive constant K. dy/ds. 1. a family of curves is obtained. z).
Stream-line. y(0). The map might be diﬀerentiable with respect to x. particle-path and streak-line Stream-line
Suppose that a velocity ﬁeld v(x. z) v(x. y. y. y.1. dz/ds). at a given time t. u(x. for such a map.5 The curve thus obtained is called a stream-line. z).e. Thus. and in addition. For the uniqueness of the solution to the initial condition.
5
Mathematically. v(x.
1. at least locally in the neighborhood of s = 0. 1. a family of stream-lines are deﬁned at each instant t (Fig.9) is assured by the continuity (and boundedness) of the three component functions of v(x. the tangent to the curve is written as (dx/ds. diﬀerentiable homeomorphism). This is written in the following way: dy dz dx = = = ds.1). existence of solutions to Eq.6
Flows
B0 would be mapped to Bt .3. v. w(x.
. z)
(1. (1. the vector ﬁeld v = (u. v. For a set of initial conditions. Namely. This kind of map is termed a diﬀeomorphism (i. t) = (u. z(0)) is determined uniquely.

then both equations are equivalent. Lagrange derivative
Suppose that the temperature ﬁeld is expressed by T (x.2). a common practice is to introduce dye or smoke at ﬁxed positions in a ﬂuid ﬂow and observe colored patterns formed in the ﬂow ﬁeld (Fig. 1. if the ﬂow ﬁeld is time-dependent (Fig. Z(t). in the way of Eulerian description.3. 1.3. Denoting the ﬁxed point P by A.e. If the ﬂow ﬁeld is steady (Fig. and they appear quite diﬀerently. 1. it is obvious that the streakline coincides with the particle-path. Thus the streak-line at a time t is represented parametrically by the function X(A. Then the particle temperature is expressed by Ta (t) = T (Xa .3).
. t) at a later time t where a τ is deﬁned by A = X(a τ . t − τ ) with the parameter τ . Consider a ﬂuid particle denoted by the parameter a in the ﬂow ﬁeld and examine how its temperature Ta (t) changes during the motion.9) and (1. t) = T (X(t). implying that both stream-lines and particle-paths are identical in steady ﬂows. t). the ﬂuid particle that has passed the point A at a previous time τ will be located at X = X(a τ . 1.4. An instantaneous curve composed of all ﬂuid elements that have passed the same particular ﬁxed point P at previous times is called the streakline. Smoke from a chimney is another example of analogous pattern. Z) and its velocity by va (t) = (ua . Streak-line
In most visualizations of ﬂows or experiments. Y (t). t) and that the velocity ﬁeld is given by v(x.3.10) are identical except the fact that the right-hand sides of (1. t). i.8
Flows
It is seen that the two equations (1. wa ).10) include the time t. Hence if the velocity ﬁeld is steady. and therefore with the streamline. τ ). then all the three lines are diﬀerent. v does not depend on t. However. Let the particle position be given by Xa (t) = (X. 1. Y.4). va .

1. Japan.2). The illumination is from upward right. 1. The smoke lines originate from equally-spaced discrete points on a vertical straight wire on the left placed at just upstream position of the cylinder (at the point of intersection of the central horizontal white line (from the left) and the vertical line connecting the two arrows out of the frame). Taneda (Kyushu University. Thus.7. and the many parallel horizontal lines in the upper and lower layers show a uniform stream of wind velocity 1 m/s from left to right. 4. S. 4. [As for the wake. This is placed in order to show how the wake reorganizes to another periodic structure of larger eddies. 1988).2. = ∂t ∂x ∂y ∂z x=Xa
. The vertical white line at the central right shows the distance 1 m from the cylinder. see Problem 4. Visualization of the wake behind a thin circular cyinder (of diameter 5 mm) by a smoke-wire method.] The photograph is provided through the courtesy of Prof. and hence the shadow line of the cylinder is visible to downward left on the lower left side. The wake is the central horizontal layer of irregular smoke pattern. and Fig. all the smoke lines are streak-lines. the time derivative of the particle temperature is given by dX ∂T dY ∂T dZ ∂T ∂T d Ta (t) = + + + dt ∂t dt ∂x dt ∂y dt ∂z ∂T ∂T ∂T ∂T + ua + va + wa = ∂t ∂x ∂y ∂z ∂ ∂ ∂ ∂ +u +v +w T . Re = 350 (see Table 4. The regular periodic pattern observed in the initial development of the wake is the K´rm´n vortex a a street. Stream-line.12).6 (Fig.3. particle-path and streak-line
9
Fig.
Hence.

The particle P started from the ﬁxed point O at a time t0 and is now located at P at t after the times t1 . and streak-lines (a thick solid line). particle-path (broken lines). 1.
It is convenient to deﬁne the diﬀerentiation on the right-hand side by using the operator. particle-path (a broken line). convective derivative or Lagrange derivative. we have dTa /dt = DT /Dt X . This derivative is called variously as the material derivative.
stream-line at t
P
The tangents of the stream-line and particle-path coincide.
th -pa cle stream-line rti pa
streak-line
Fig. Dt ∂t ∂x ∂y ∂z which is called the convective derivative.10
Flows
stream-line
S •
partic
le-pa
th
streak-line dye
Fig.
The tangents of the stream-line and streak-line coincide. ∂x := ∂/∂x. ∂ ∂ ∂ ∂ D := +u +v +w = ∂t + u∂x + v∂y + w∂z . this time derivative denotes the diﬀerentiation following the particle motion.3.
a
. t2 and t3 . Thus. Unsteady ﬂow: stream-lines (thin solid lines). and so on. As is evident from the above derivation. and streak-line (a thick solid line). Steady ﬂow: stream-lines (thin solid lines). where ∂t := ∂/∂t. 1.4.

we consider a relative motion of ﬂuid in a neighborhood of an arbitrarily chosen point P = x = (x1 . Decomposition
In order to represent such a local motion mathematically.15) (1. Using ∇ and v = (u. where the velocity is v = (v1 . etc. where f (x.1. Relative motion (1. each ﬂuid element moves subject to straining deformation and local rotation. v3 ).13)
where the dot denotes the inner product [Appendix A. one can write D = ∂t + u∂x + v∂y + w∂z = ∂t + v · ∇. This is shown as follows.4. grad f = ( ∂x .2 and see (A. we have DQ = 0. z) is a diﬀerentiable scalar function. Dt 1.14)
Given a velocity ﬁeld v(x. x3 ). Dt If the value is invariant during the particle motion. 1. Writing the velocity of a neighboring point Q = x + s as v + δv at
.4. ∇ = ( ∂x . and the vector gradf is called the gradient of the function f (x). temperature T . The diﬀerential operator ∇ with three components is termed the nabla operator. Relative motion
11
Let us introduce the following diﬀerential operators. y. t). ∂z ).4.1. ∂z ) f. v2 . ∂y . x2 . Suppose that we have a physical (scalar) ﬁeld Q(x.7)]. v. Its time derivative following the motion of a ﬂuid particle is given by DQ = ∂t Q + (v · ∇)Q. Dt (1. ∂y . t) such as density ρ. w).

The ink covers a certain compact area of the surface. This pattern is a snap-shot at an instant and consists of a number of curves. V dt ∂x ∂y ∂z (1.31)
. ∂vx ∂vy ∂vz 1 dV = div v = + + .
1.2 Divergence operator div Consider a small volume of ﬂuid of a rectangular parallelepiped in a ﬂow ﬁeld of ﬂuid velocity v = (vx . particle-paths or streak-lines. some ink pattern will be observed. it is found that δv (a) represents local rigid-body rotation.5. vy . it is independent of the 2 displacement vector s.5). What sort of lines are the curves printed on the paper? Are they stream-lines. and the water is set in motion and its horizontal surface is in smooth motion. it is found that the local relative velocity δv consists of a pure straining motion δv (s) and a local rigid-body rotation δv (a) . After a while. vz ). a pattern will be printed on the paper. or other kind of lines?
Problem 1. every point s in the neighborhood of x rotates with the same angular velocity. The ﬂuid volume V changes under the straining motion. Problems
15
This component of relative velocity describes a rotation of the angular velocity 1 ω. Although ω depends on x.5.1 Pattern of ink-drift Suppose that some amount of water is contained in a vessel. which is called the ink-drift printing (Fig. If a sheet of plain paper (for calligraphy) is placed quietly on the free surface of the water. 1. Let a liquid-drop of Chinese ink be placed quietly on the ﬂat horizontal surface maintaining a ﬂow with some eddies.
Problems
Problem 1. In summary.1. Namely. Show that the time-rate of change of volume V per unit volume is given by the following. Thus.

). are regarded as continuous and diﬀerentiable functions of position and time. but state variables vary from point to point. all variables. entropy.1. or internal energy. momentum and energy.2) can be applied. enthalpy. energy and thermodynamic variables (pressure. such as mass density. viscosity and thermal conductivity in equations of motion. there occurs exchange of physical quantities dynamically and thermodynamically. temperature. each part of the material is assumed to be in equilibrium mechanically and thermodynamically with the surrounding medium. Equilibrium in a material is represented by the property that the thermodynamic state-variables take uniform values at all points of the material. In the continuum representation of ﬂuids. macroscopic motions of the matter (a ﬂuid) are less sensitive to whether the structure of matter is discrete or continuous. In this situation. in most circumstances where real ﬂuids are exposed. momentum. This is considered to be due to the molecular structure or
17
. In ﬂuid mechanics. However. the ﬂuids are hardly in equilibrium.
Continuum and transport phenomena
The motion of a ﬂuid is studied on the basis of the fundamental principle of mechanics. When the state variables are not uniform. the matter is brought to an equilibrium in most circumstances by the exchange. Usually when external forcing is absent. namely the conservation laws of mass. the eﬀect of actual discrete molecular motion is taken into account as transport phenomena such as diﬀusion. For a state of matter to which the continuum hypothesis (Sec. etc. 1.Chapter 2
Fluids
2.

This type of exchange is called the transfer phenomenon. and the exchange is understood as the transfer.
2. Therefore. Suppose that the concentration of one constituent β of matter is denoted by C which is the mass proportion of the component with respect to the total mass ρ in a unit volume. The entropy law is a typical one in this regard. and C is assumed to be a diﬀerentiable function of point x and time t: C(x.1). (2. there is a net transfer as a balance of the two counter ﬂuxes (Fig. Within the mixture.1. t). Let us write the net transfer through δA(n) toward the direction n per unit time as q(x) · n δA(n).
Two counter ﬂuxes. Hence. the mass density of the component is given by ρC.2. we have three transfer phenomena: mass diﬀusion.
Mass diﬀusion in a ﬂuid mixture
Diﬀusion in a ﬂuid mixture occurs when composition varies with position. Diﬀusion of the component β through the surface δA occurs from one side to the other and vice versa. we choose an arbitrary surface element δA with its unit normal n. 2. For conservative quantities. momentum diﬀusion and thermal diﬀusion.
.18
Fluids
due to random interacting motion of uncountably many molecules.1)
n C (x2)
C (x1)
Fig. However. it is possible to connect the decrease of some quantity at a point to the increase of the same quantity at another point. owing to the nonuniformity of the distribution C(x). corresponding to the three conservation laws mentioned above. with conservative variables. 2. the exchange of variables make perfect sense. or transport phenomenon. Because.

a macroscopic scale is much larger than the microscopic intermolecular distance. The total mass of the component β in the volume V is Mβ = V ρC dV by the deﬁnition of C. ∂z C). there is no macroscopic motion. Some of this component will move out of V
A n
V
Fig. (2. ∂y C. 2. the diﬀusion ﬂux q will be related to the concentration C. the ﬂux q would depend on the gradient linearly with the proportional constant kC as follows: q(C) = −kC grad C = −kC (∂x C. So that. so that the concentration gradient in usual macroscopic problems would be very small from the view point of the molecular structure. In an anisotropic medium. Mass diﬀusion in a ﬂuid mixture
19
assuming that it is proportional to the area δA.
. and observe the volume V with respect to the frame of the center of mass. the coeﬃcient should be a tensor kij . and a vector q can be deﬁned at each point x. From the aspect of molecular motion. where q is called the diﬀusion ﬂux. Provided that the concentration gradient is small.
An arbitrary volume V . resulting in attenuation of the degree of C nonuniformity. rather than a scalar constant kC . The above expression (2. we choose an arbitrary volume V bounded by a closed surface A in a ﬂuid mixture at rest (Fig. Since the ﬂux q should be zero if the concentration is uniform.2.2.2)
where kC is the coeﬃcient of mass diﬀusion.2. The coeﬃcient kC is positive usually.2) is valid in an isotropic material. in order to derive an equation governing C.2). 2. From the above consideration. This is regarded as a mathematical assumption that higher-order terms are negligible when the ﬂux is represented by a Taylor series with respect to derivatives of C. Next. q would depend on the concentration gradient or derivatives of C. and the diﬀusion ﬂux is directed from the points of larger C to those of smaller C.

6) transforming the surface integral into a volume integral. ∂t (2. no net translation of mass is possible. ρ (2. In this case. Hence we have the equation for the rate of increase of Mβ : d dt
V
ρC dV =
V
∂ ρC dV = ∂t
A
kC n · ∇C dA. ∂x2 ∂y ∂z λC = kC . Thus we obtain ∂ (ρC) = div(kC ∇C). ∂t where ∆ is the Laplacian operator.4)
1
When the diﬀusing component is only a small fraction of total mass. we obtain ∂ (ρC) − div(kC ∇C) dV = 0. the integrand must vanish identically.
. ∂t
V
Since this relation is valid for any volume V . Therefore. the diﬀusion coeﬃcient kC is assumed to be constant. the total amount of outward diﬀusion is given by
A
q(C) · n dA = −
A
kC n · grad C dA.20
Fluids
by the diﬀusion ﬂux q through the bounding surface A. the density ρ may be regarded as constant even when the frame is not of the center of mass. ∆ := ∇2 = ∂2 ∂2 ∂2 + 2 + 2. Applying the Gauss’s theorem (see Sec.3)
In a ﬂuid at rest in equilibrium.
where n is unit outward normal to the surface element dA. This outward ﬂux gives the rate of decrease of the mass Mβ (per unit time). the above equation reduces to ∂C = λC ∆C.
where the time derivative is placed within the integral sign since the volume element dV is ﬁxed in space. 3. and grad is replaced by ∇. the total mass ρ in unit volume is kept constant.1 Moreover.1 and Appendix A.

A molecule in a gas carries its own kinetic energy. where Cp is the speciﬁc heat per unit mass at constant pressure. The average kinetic energy of molecular random velocities is the thermal energy. and those vice versa. where the vector q is now called the heat ﬂux.e.4) is the diﬀusion equation. then the transfer of thermal energy (from one side to the other) cancels out with the counter transfer. The ﬂow of heat through the surface element δA will be written in the form (2.3. heat transport is caused by collision or interaction between neighboring molecules.3) is written as ρCp ∂T = div(k∇T ).
Thermal diﬀusion
Transport of the molecular random kinetic energy (i.2. In liquids or solids.
2. Analogously with the mass diﬀusion.2). However. The equation corresponding to (2.1). (2.3. The second law of thermodynamics (for the entropy) implies that the coeﬃcient k should be positive (see Sec. and there is no net heat transfer. If the temperatures on both sides are equal. if the temperature T depends on position x. obviously there is a net heat transfer.
. Thermal diﬀusion
21
Equation (2. Choosing an imaginary surface element δA in a gas. the heat ﬂux will be represented in terms of the temperature gradient as q = −k grad T. and λC is the diﬀusion coeﬃcient. we consider such molecules moving from one side to the other. 1. the heat energy) is called heat transfer. ∂t
since the increase of heat energy is given by ρCp ∆T for a temperature increase ∆T . which deﬁnes the temperature T (Sec.5)
where k is termed the thermal conductivity. 4.2).

3. Equation (2.7)
Fig. Let us consider the transport of the ith component of momentum. ρcp (2. 1.
Momentum transfer through a surface element δA(n).6)
in a ﬂuid at rest. A ﬂuid with such an internal friction is said to be viscous. The concentration and temperature considered above were scalars. the ith momentum transfer through a surface element δA(n) from the side I (to which the normal n is directed) to the other II is deﬁned (Fig.22
Fluids
Corresponding to (2.
Momentum transfer
Transfer of molecular momentum emerges as an internal friction. 2.4. Constituent molecules in the ﬂuid particle are mov˜ ing randomly with velocities uα (Sec.
.2).1).
2. where λT is the thermal diﬀusivity. The momentum transfer is caused by molecules carrying their momenta. Macroscopic velocity v of a ﬂuid at a point x in space is deﬁned by the velocity of the center of mass of a ﬂuid particle located at x instantaneously. the equation of thermal conduction is given by ∂T = λT ∆ T. ∂t λT = k . 2. Instead of the expression (2.3) as qij nj δA(n).4).6) is also called the Fourier’s equation of thermal conduction. This requires some modiﬁcation in the formulation of momentum transfer. (2. or by interacting force between molecules. however momentum is a vector.

The dimension of qij is equivalent to that of force per unit area. there is another signiﬁcant diﬀerence from the previous cases of the transfer of concentration or temperature. both stresses counter balance. (2. In a ﬂuid at rest. Hence. resulting in vanishing net ﬂux in the equilibrium. However. hence qij nj = −pni . both are same and we have twice the positive momentum gain P (stress). In the case of heat. Let us pay attention to a neighborhood on one side of a surface element δA(n) where the normal vector n is directed.4.8)
(see Eq. while the positive momentum coming into the side I through δA would be expressed as “emerging of positive momentum” P which is a contraposition of the previous statement. 2. variables are distributed uniformly in space. 4. and called the viscous stress. on the other side of the surface δA. The stress associated with nonuniform velocity ﬁeld v(x) is characterized by a tangential force-component to the surface element considered. the momentum transfer is given by qij = −pδij . Momentum transfer
23
where qij nj = 3 qij nj .4). the pressure is always normal to the surface chosen (Fig. Total pressure force acting
. Concerning the momentum transfer.1 for δij ).1 and Appendix A. How about in the case of momentum? The negative momentum (because it is anti-parallel to n) escaping from from the side I out of δA would be expressed as “vanishing of negative momentum” Q. the heat ﬂux escaping out of δA is balanced with the ﬂux coming in through δA. The tensor quantity qij represents the j=1 ith component of momentum passing per unit time through a unit area normal to the jth axis. and such a quantity is termed a stress tensor. This is recognized as the pressure. Thus.1)). (4. It can be veriﬁed that the stress tensor must be symmetric (Problem 2. where δij is the Kronecker’s delta and the minus sign is due to the deﬁnition of qij (see the footnote to Sec. In a uniform ﬂuid.2. the situation is reversed and we have twice the loss of P . Suppose that the ﬂuid is at rest and is in both mechanical and thermodynamical equilibrium.3): qij = qji . Hence.

Suppose there is a parallel ﬂow along a ﬂat plate with the velocity far from it being U in the x direction. The ﬂow ﬁeld represented as (u(y).24
Fluids
Fig.5. termed as a boundary layer. (2. 0) in the (x. the y axis being taken perpendicular to the plate. In the transport phenomena considered above such as diﬀusion of mass.9)
Sp
where Sp denotes the surface of a small ﬂuid particle. Owing to this shear ﬂow.4.
on a ﬂuid particle is given by − pni dA. the net transfers are in the direction of diminishing nonuniformity (Sec. If the friction
. 4.
Pressure stress. 0) is called a parallel shear ﬂow. If the velocity far from the wall is large enough. heat or momentum. thermal conductivity and viscosity in the representation of ﬂuxes are called the transport coeﬃcients. the proﬁle of the tangential velocity distribution perpendicular to the surface has a characteristic form of a thin layer. and the ﬂow velocity is represented by (u(y). An ideal ﬂuid and Newtonian viscous ﬂuid
The ﬂow of a viscous ﬂuid along a smooth solid wall at rest is characterized by the property that the velocity vanishes at the wall surface. The coeﬃcients of diﬀusivity. y) cartesian coordinate frame. 2. the plate is acted on by a friction force due to the ﬂow. 2.2).

This law can be extended to the law on an internal imaginary surface of the ﬂow. Consider an internal surface element B perpendicular to the y-axis located at an arbitrary y position (Fig.7). 2.1). The pressure stress has only the normal component to the surface δA(n).
. If the friction σ (s) per unit area is written as σ (s) = µ d u(y). in particular.0) y B
Fig. whereas the viscous stress has a tangential component and a normal component (in general compressible case). In the ﬂow of an inviscid ﬂuid.7) and (2. and it corresponds to qxy of (2.5. called the shear stress for the present shear ﬂow. The unit normal to the surface B is in the positive y direction. and the ﬂuid has nonzero tangential
u(y) n = (0. dy (= qxy ).8) in the previous section is another surface force.
Momentum transfer through an internal surface B. this is called the Newton’s law of viscous friction (see Problem 2. The internal friction force on B from the upper to lower side has only the x-component. The friction σ (s) per unit area is called the viscous stress.2.11)
then the ﬂuid is called the Newtonian ﬂuid. Such a ﬂuid is called an inviscid ﬂuid.10)
where µ is the coeﬃcient of shear viscosity. The stress is also termed as a surface force.5). or an ideal ﬂuid. One can consider an idealized ﬂuid in which the shear viscosity µ vanishes everywhere. An ideal ﬂuid and Newtonian viscous ﬂuid
25
force per unit area of the plate is represented as σf = µ du dy
y=0
. The pressure force given by (2. (2.5.1. the velocity adjacent to the solid wall does not vanish in general.
(2. 2.

(2. 2.6.13)
The inviscid ﬂuid is often called an ideal ﬂuid (or sometimes a perfect ﬂuid). Dii = eii − 1 1 3 Dδii = D − 3 D · 3 = 0 since δii = 3 . the viscous stress is given in general by σij = 2µDij + ζDδij . Viscous stress
For a Newtonian ﬂuid. In fact. the ideal ﬂuid denotes a ﬂuid characterized by the property that all transport coeﬃcients of viscosity and thermal conductivity vanish. the boundary conditions of the velocity v on the surface of a body at rest are summarized as follows: Viscous ﬂuid: v = 0 Inviscid ﬂuid: nonzero tangential velocity (no-slip). where µ and ζ are coeﬃcients of viscosity.15) (2. and 1 1 1 Dij := eij − Dδij = (∂i vj + ∂j vi ) − (∂k vk ) δij 3 2 3 D := ∂k vk = ekk = div v. the ﬂow velocity of a viscous ﬂuid vanishes at the solid wall. It may be said that Dij is a deformation tensor associated with a straining motion which keeps the volume unchanged. This is termed as no-slip.12) (2.
. On the other hand. Thus. In this textbook.14)
The tensor Dij is readily shown to be traceless.26
Fluids
slip-velocity at the wall.14) of the viscous stress (v) can be derived from a general linear relation between the stress σij and the rate-of-strain tensor eij for an isotropic ﬂuid. (2.16)
(v)
(2. in which the surface force has only normal component. (slip-ﬂow). in which the number of independent scalar coeﬃcients is only two — µ and ζ (see Problem 2. macroscopic ﬂows of an ideal ﬂuid is separated from the microscopic irreversible dissipative eﬀect arising from atomic thermal motion. Since all the transport coeﬃcients vanish. The expression (2.4).

y. b. Problem 2.
(v)
. c are constants. (2.4 Symmetry of stress tensor Suppose that a stress tensor σij is acting on the surface of an inﬁnitesimal cubic volume of side a (with its edges parallel to the axes x. z) in a ﬂuid of density ρ. derive the expression (2. where ekl is deﬁned by (1.5 Stress and strain Consider a ﬂow ﬁeld vk (x) of an isotropic viscous ﬂuid. Using this form and the symmetry of the stress tensor (Problem 2.14) for the (v) viscous stress σij : σij = 2µDij + ζDδij .3).18). In an isotropic ﬂuid.27)
(v)
where a. and suppose (v) that there is a general linear relation between the viscous stress σij and the rate-of-strain tensor ekl : σij = Aijkl ekl . Considering the balance equation of angular momentum for the cube and taking the limit as a → 0. verify that the stress tensor must be symmetric: σij = σji .30
Fluids
Problem 2. the coeﬃcients Aijkl of the fourth-order tensor are represented in terms of isotropic tensors (see (2. which are given as follows: Aijkl = a δij δkl + b δik δjl + c δil δjk .8)).

Fluid particles move about in the space with a velocity dx/dt = v(x. which are conservation of mass. or [LL75]. the conservation of momentum is derived. The ﬁeld variables denote their values at a point x and at a time t. t). and so on. From the homogeneity of space. p. these conservation laws are represented in terms of ﬁeld variables such as v. t).1).Chapter 3
Fundamental equations of ideal ﬂuids
Fluid ﬂows are represented by ﬁelds such as the velocity ﬁeld v(x. t). 31
. density ﬁeld ρ(x. or equivalently (x1 . Conservation of mass results from the invariance of the relativistic Lagrangian in the Newtonian limit (Appendix F. y. pressure ﬁeld p(x. Conservation of angular momentum (which is not discussed in this chapter) results from isotropy of space. There are three kinds of conservation laws of mechancis. ρ. See Chap. y. t). etc. Flow ﬁeld evolves with time according to fundamental conservation laws of physics. x2 . momentum and energy. t). Since the ﬁeld variables depend on (x. The position vector x is represented by (x. [LL76].
1
The conservation laws in mechanics result from the fundamental homogeneity and isotropy of space and time. z). temperature ﬁeld T (x. From the homogeneity of time in the Lagrangian function of a closed system.1 In ﬂuid mechanics. x3 ) in the cartesian frame of reference. the conservation of energy is derived. z) and t. the governing equations are of the form of partial diﬀerential equations. 12.

(3.
Mass conservation
The law of mass conservation is represented by the following Euler’s equation of continuity. M0 (t) =
V0
ρdV. t) = (u.1. 3.
Volume V0 . ∂t ∂x ∂y ∂z (3.1). Fluid mass in the volume dV is given by ρdV .1)
where ρ is the ﬂuid density and v(x. the ﬂuid itself moves around with velocity v.3)
The ﬂuid density ρ depends on t.
. this is written as ∂t ρ + div(ρv) = 0. the time deriative ∂t ρdV can be replaced by (∂t ρ)dV . and choose a volume element dV within V0 .32
Fundamental equations of ideal ﬂuids
3. Since we are considering ﬁxed volume elements dV in space. Therefore.2)
and derived in the following way. and its rate of change is given by ∂ d M0 (t) = dt ∂t
V0
ρdV =
V0
∂ρ dV.4)
where the partial diﬀerential operator ∂/∂t is used on the right-hand side. which reads ∂ρ ∂(ρu) ∂(ρv) ∂(ρw) + + + = 0.
Fig. and the total mass is its integral over the volume V0 . the total mass M0 varies with time t. ∂t
(3. Take a certain volume V0 ﬁxed in space arbitrarily (Fig.1. v. and in addition.
(3. Using the diﬀerential operator div of the vector analysis. 3. w) the velocity.

so that ρvn dA denotes the mass of ﬂuid ﬂowing out of the volume V0 per unit time. that is equal to −dM0 /dt. where vn is the normal component of the velocity2 and vn dA denotes the volume of ﬂuid passing through dA per unit time (Fig.3.
This integral over the closed surface A0 is transformed into a volume integral by Gauss’s divergence theorem3 :
A0
ρvk nk dA =
V0
∂ (ρvk ) dV = ∂xk
div(ρv) dV. The amount of ﬂuid ﬂowing through a surface element dA with the unit normal n is given by ρvn dA = ρv · ndA.
vn = v · n = vk nk = |v||n| cos θ.
2
Mass ﬂux through dA. Total out-ﬂow of the ﬂuid mass per unit time is
A0
ρ vn dA =
A0
ρv · ndA =
A0
ρvk nk dA.2). 3. 3 The rule for transforming the surface integral into a volume integral with the bounding surface is as follows: the term nk dA in the surface integral is replaced by the volume element dV and the diﬀerential operator ∂/∂xk acting on the remaining factor in the integrand of the surface integral. 3.
V0
(3. Mass conservation
33
The change of total mass M0 is caused by inﬂow or outﬂow of ﬂuid through surface A0 bounding V0 . on
Fig. where |n| = 1 and θ is the angle between v and n.31).2.5)
where the div operator is deﬁned by (1.
. The normal n is taken to be directed outward from V0 .1. This gives the rate of decrease of total mass in V0 . Thus.

4) and (3. ∂xk where ∂k = ∂/∂xk .9) is also obtained from (3. Eq. Hence. the ﬂuid is called incompressible. we have div v = ∂x u + ∂y v + ∂z w = 0.34
Fundamental equations of ideal ﬂuids
addition of the two terms. the right-hand sides of (3. ∂t ∂xk The second term of (3. In this case.2). we obtain the equation of continuity (3. the integrand inside [ ] must vanish pointwise.7)
Notes: (i) If the density of each ﬂuid particle is invariant during the motion.7) is rewritten as Dρ + ρ div v = 0. (3. e. the integral does not always vanish. then Dρ/Dt = 0.13).
4 Otherwise. the continuity equation (3.g.8) (3. Using the Lagrange derivative D/Dt of (1. must vanish: ∂ρ dV + ∂t div(ρv)dV =
V0 V0
V0
∂ρ + div(ρv) dV = 0.7) is decomposed as ∂ (ρvk ) = vk ∂k ρ + ρ∂k vk = (v · ∇)ρ + ρ div v. ∂t
(3. Dt (3. (ii) Uniform density: The same equation (3. If the ﬂuid is incompressible.
.9)
This is valid even when ρ is not uniformly constant. which is also written as ∂ ∂ ρ+ (ρvk ) = 0.2) by setting ρ to be a constant.6)
This is an identity and must hold for any choice of volume V0 within the ﬂuid. Thus. when choosing V0 where the integrand [ ] is not zero.9) implies both cases (i) and (ii).4 Thus. (3.5).

3.3. (3.7).3.5
Fig.10) is called the conservation form. (ρδV )(acceleration) = (force).10) reduces to the continuity equation (3. This conservation law results from the homogeneity of space with respect to the Lagrangian function in Newtonian mechanics. Fundamental equations and conservation equations for continuous ﬁelds is the subject of Chapter 12 (gauge principle for ﬂows of ideal ﬂuids).
.3).2. The equation of the form (3.
Conservation form
It would be instructive to remark a general characteristic feature of Eq. then Eq. Namely. we write down ﬁrstly the equation of motion for a ﬂuid particle of mass ρδV in the form. Momentum conservation
The conservation of momentum is the fundamental law of mechanics.
Conservation form. and Q is a source generating the ﬁeld D (Fig.3. 3. If D is the mass density ρ and (F)k is the mass ﬂux ρvk passing through a unit surface per unit time (and there is no mass source Q = 0). 3.3. (3. in general.
5
This relation is the Newton equation of motion itself for a point mass (a discrete object). Conservation form
35
3. However.7).10)
k
where D is a density of some physical ﬁeld and (F)k is the kth component of corresponding ﬂux F. it has the following structure: ∂ ∂ D+ F ∂t ∂xk = Q. (3.

the surface force is only the pressure (2.33)). It is remarkable to ﬁnd that the force on a small ﬂuid particle of volume δV is given by the pressure gradient. −grad p δV = −(∂x p. (3. (ρδV ) g with g as the acceleration of gravity. By this extended Gauss theorem.1. we obtain 1 ∂t v + (v · ∇)v = − grad p + f .36
Fundamental equations of ideal ﬂuids
Thereafter. Equation of motion
In an ideal ﬂuid. ρ (3. there is usually a volume force which is proportional to the mass ρδV . 3. In addition to the surface force just given.33). Dt (3.1. the total pressure force on A0 is given by − pni dA = − ∂ pdV = − ∂xi (grad p)i dV. ∂z p) δV.13)
. 3.9) (there is no viscous stress). A typical example is the gravity force. Choosing a volume V0 as before and denoting its bounding surface by A0 .8) or (2.11)
A0
V0
V0
where the middle portion is obtained by applying the rule described in the footnote in Sec.12)
Dividing this by ρδV and using the expression given in (1. The Newton equation of motion for a ﬂuid particle is written as (ρδV ) D v = −grad pδV + (ρδV) f . the surface integral is transformed to the volume integral. which is often called an external force. the acceleration of the ﬂuid particle is written as Dv/Dt (see (1.3. ∂y p. t) is given. it will be shown that this is equivalent to the conservation of momentum. Let us write this as (ρδV ) f . Once the velocity ﬁeld v(x.

(3.29) of Chap. ∂t v + u∂x v + v∂y v + w∂z v = −ρ−1 ∂y p. 3.14)
If the external force is the uniform gravity represented by f = (0.
. 5.3. the ith component of the equation is 1 ∂t vi + vk ∂k vi = − ∂i p + fi . 0.1). Lagrange. It is rather surprising to ﬁnd that not only Euler (1757) derived an integral for potential ﬂows of the form (5. 2
(3. The nonlinearity is responsible for complex behaviors of ﬂows. To see it.6 The second term (v · ∇)v on the left-hand side is also written as (v · grad)v. Momentum conservation
37
This is called Euler’s equation of motion. Before that. This is of the second order with respect to the velocity vector v.1 was derived in 1752. and often called a nonlinear term.3. 1752) by assuming that continuity of the ﬂuid is never interrupted.21) is useful: v × (∇ × v) = ∇
6
1 2 v − (v · ∇)v. carried out by Bernoulli family. each component is written down as follows: ∂t u + u∂x u + v∂y u + w∂z u = −ρ−1 ∂x p. in the 18th century [Dar05]. Principes g´n´raux du e e mouvement des ﬂuides (General principles of the motions of ﬂuids). the following vector identity (A. he showed in the same paper the equation of continuity exactly of the form (3.15)
Euler’s equation of motion is given another form. determined from the principles of mechanics (as carried out in the main text). d’Alembert. ∂t w + u∂x w + v∂y w + w∂z w = −ρ−1 ∂z p − g. his contribution should be regarded as one step toward the end of a long process. etc. the term ∂t v is related to ﬂuid inertia as well).15) (with x.32) of Sec. The form of equation Euler actually wrote down is that using components such as (3.4. ρ (3. In the component representation. This term is also called an inertia term (however. Thus. or advection term.16)
The paper of Leonhard Euler (1707–1783) was published in the proceedings of the Royal Academy Prussia (1757) in Berlin with the title. Euler presented an essential part of modern ﬂuid dynamics of ideal ﬂuids. The equation of continuity had been derived in his earlier paper (Principles of the motions of ﬂuids. but also the z-component of the vorticity equation (3. y components of external force). −g) where the acceleration of gravity g is constant and directed towards the negative z axis. However.

replace ∂k (X) dV with Xnk dA. (3. which is seen to have the form of Eq. Momentum conservation
39
where the tensor Pik is deﬁned by Pik = ρvi vk + pδik . the Euler equation of motion represents the momentum conservation in conjunction with the continuity equation. i. Moving this term to the right-hand side.23)
V0
A0
V0
This is interpreted as follows. whereas the momentum density is a vector.
. while Pik denotes the momentum ﬂux (tensor).
7
The mass density was a scalar.3.
V0
V0
V0
The second integral term on the left-hand side can be transformed to a surface integral by using the rule of the Gauss theorem. whereas the ﬁrst integral on the right-hand side represents the momentum ﬂowing out of the bounding surface A0 per unit time. (3. (3. Correspondingly.21) represents the conservation of momentum. This is natural because the external force is a source of momentum in the true meaning. In fact. The left-hand side denotes the rate of change of total momentum included in volume V0 .7 The diﬀerence from the previous equation of mass conservation is that there is a source term ρfi on the righthand side.10).21) a source (production) of momentum will become clearer. we have ∂ ∂t ρvi dV + ∂ Pik dV = ∂xk ρfi dV. if we integrate the equation over a volume V0 . we obtain ∂ ∂t ρvi dV = − Pik nk dA + ρfi dV. and the second integral is the total force acting on V0 which is nothing but the rate of production of momentum per unit time in mechanics.22)
Equation (3.3. Why is the term on the right-hand side of (3. the ﬂux Pik is a second-order tensor. The quantity ρvi is the ith component of the momentum density (vector).e. Thus.

whereas the second represents a microscopic momentum ﬂux. Then the equations will be simpliﬁed. Denoting the entropy per unit mass by s.22) is used. Adiabatic motion
In an ideal ﬂuid. we have a
. the motion of a ﬂuid particle is adiabatic.2 we saw that the microscopic momentum ﬂux gives the expression for pressure of an ideal gas. there is neither heat generation by viscosity nor heat exchange between neighboring ﬂuid particles. Energy conservation
The conservation of energy is a fundamental law of physics. 3. This equation is transformed to ∂t (ρs) + div(ρsv) = 0. and the entropy of each ﬂuid particle is invariant during its motion. 3. Because of this property. Dt (3. 1. Denoting the internal energy and enthalpy per unit mass by e and h.2). Pik nk dA = ρvi vk nk dA + pδik nk dA. This is called a homentropic ﬂow. The ﬁrst term is the macroscopic momentum ﬂux.1. the motion is isentropic.4. This represents the ith component of total momentum ﬂux passing through the surface element dA. respectively and the speciﬁc volume by V = 1/ρ.40
Fundamental equations of ideal ﬂuids
An important point brought to light is the following expression. the value is invariant thereafter. the adiabatic motion of a ﬂuid particle is described as D s = ∂t s + (v · ∇)s = 0. (3. It will be shown that this is consistent with the conservation of energy. In Sec.24)
where (3. (3.4.26)
by using the continuity equation (3. An ideal ﬂuid is characterized by the absence of viscosity and thermal diffusivity.25)
It is said. If the entropy value was uniform initially. Therefore the motion of an ideal ﬂuid is adiabatic.

(3. and energy ﬂux Qn = ρEvn + pvn =
. we have ∂t 1 2 ρv + ρe dV = − 2 +
V0
V0
A0
1 2 ρv + ρe + p vk nk dA 2 W dV.3. Integrating (3.5. in addition to the rate of work W by the external force.
energy density pvn energy flux
Fig. 3. we ﬁnally obtain ∂t ρ 1 2 v +e 2 + ∂k ρvk 1 2 v +h 2 = W.38)
This indicates that the rate of increase of the total energy in V0 is given by the sum of the energy inﬂow − 1 ρv 2 + ρe vk nk through 2 the bounding surface and the rate of work by the pressure −pvk nk .4). It is remarked that the energy density is given by the sum of kinetic energy and internal energies.10).37)
where W = ρvk fk denotes the rate of work by the external force. Energy conservation
43
Using the entropy equation (3.4. This is the equation of conservation of energy in the form of (3.25). 2
namely e is replaced by h for the ﬂux F . (3. whereas the energy ﬂux (F)k is (F)k = ρvk 1 2 v +h .
1 2 v 2
+ e.37) over volume V0 and transforming the volume integral of the second term into the surface integral. When there is no external force. 3. the right-hand side W vanishes (Fig. Energy density E = 1 ρ( 2 v 2 + h)vn .

13).44
Fundamental equations of ideal ﬂuids
3. t). (3. 0). Then.1 One-dimensional unsteady ﬂow Write down the three conservation equations of mass. we obtain ∂p ∂X ∂X =− +f · .5. velocity is v = (u(x. t) = −∂x p + fx . corresponding to the Eulerian version (3. Similarly. the continuity equation for the particle density ρa (t) is given by ∂t2 X · ρa (t) ∂(x) = ρa (0).
.12) can be written as ∂t2 X(a. the x component of equation of motion (3. ∂(a) (3.
Problems
Problem 3. We multiply this by ∂x/∂a. t) in (1.12) and summing up the three expressions. 0. and so on.39) ∂a ∂a ∂a This is the Lagrange’s form of equation of motion (1788). t).40)
where ∂(x)/∂(a) =det(∂Xj /∂ai ) is the Jacobian determinant.11) with its velocity Va (t) = ∂t X(a. t). momentum and energy for one-dimensional unsteady ﬂows (in the absence of external force) with x as the spatial coordinate and t the time when the density is ρ(x. multiplying y and z components of Eq. Comment: Lagrange’s form of equation of motion Lagrangian representation of position of a ﬂuid particle a = (a. (3. c) at time t is deﬁned by X(a. Deﬁning the Lagrangian coordinates a = (ai ) by the particle position at t = 0. b.

22). (2.2)
where Pik = ρvi vk + pδik is the momentum ﬂux tensor for an idealﬂuid ﬂow.1. The conservation of momentum of ﬂows of an ideal ﬂuid is given by (3. a viscous stress should be added to the “ideal” momentum ﬂux Pik .5.8). A typical stress is the pressure pδij . Another stress was (v) the viscous stress (internal friction) σij considered in Sec. the transformation δij Aj is Ai .1 which is written as   p 0 0 (p) (4.9)). 2. The surface force is also termed as the stress and represented by a tensor.21) and (3. For any vector Ai . A typical one is the shear stress in the parallel shear ﬂow in Sec. (4. 2.1) σij = −pδij = −  0 p 0  . In order to obtain the equation of motion of a viscous ﬂuid.
Equation of motion of a viscous ﬂuid
In the previous chapters. 0 0 p The pressure force acts perpendicularly to a surface element δA(n) and is represented as −pni dA (see (2. which reads ∂t (ρvi ) + ∂k Pik = ρfi .
The tensor of the form δij is called an isotropic tensor. we learned the existence of surface forces to describe ﬂuid motion in addition to the volume force such as gravity. which has a tangential force to the surface δA(n) as well.Chapter 4
Viscous ﬂuids
4. 45
1
.6.

8) is also called the Navier–Stokes equation.4. The viscous stress in an incompressible ﬂuid (D = div v = 0) is characterized by a single viscosity µ.
(v)
(4. σik = µ (∂k vi + ∂i vk ).5 0. where the external force f is added on the right-hand side. by Poisson in 1829. and by Stokes in 1845. If the external force is written as f = −∇χ (i.99 0. . the equation of motion is ∂t v + (v · ∇)v = − 1 ∇p + ν∇2 v + f ρ0 (4.8 1. by Saint-Venent in 1837. [Dar05]. µ [g/(cm · s)] Air Water Glycerin Olive oil Mercury 0. so that div v = 0. Each had his own way to justify the equation .011 6.
. . Dynamic viscosity µ and kinematic viscosity ν = µ/ρ0 at temperature 15◦ C (1 atm).1. Denoting the uniform density by ρ0 . Equation (4.9)
Table 4.15 0. by Navier in 1821.0012
4
The Navier–Stokes equation was discovered and rederived at least ﬁve times.18 · 10−3 0. conservative) where χ = gz with g the constant acceleration of gravity. then the
Table 4. Each new discoverer either ignored or denigrated his predecessors’ contribution. while µ itself is called the dynamic viscosity.8)
together with div v = 0.4 The equation becomes considerably simpler if the ﬂuid is regarded as incompressible. by Cauchy in 1823.1. The coeﬃcient ν = µ/ρ0 is termed the kinematic viscosity. Equation of motion of a viscous ﬂuid
47
This is called the Navier–Stokes equation.e.08 0.16 · 10−1 ν [cm2 /s] 0.1 shows the dynamic viscosity µ and kinematic viscosity ν of various ﬂuids.11 · 10−1 8.

The ﬁrst two terms are the rate of entropy production due to internal friction. Namely. while the last term is that owing to thermal conduction.5). Sec. The entropy can only increase.
(v)
(4. 49]. The expression on the left ρT (Ds/Dt) is the quantity of heat gained per unit volume in unit time.17)
[LL87. Energy dissipation in an incompressible ﬂuid
In an incompressible viscous ﬂuid of uniform density ρ0 where div v = 0. (4. one has ρ0 (vi ∂t vi + vi vk ∂k vi ) = −vi ∂i p + vi ∂k σik . Energy dissipation in an incompressible ﬂuid
49
hold [LL87. and
1 = ∂k vk v 2 2
.3. In the entropy equation (3. one can derive the following equation: d dt
V0
ρsdV =
µ (∂k vi + ∂i vk − (2/3)(div v)δik )2 dV 2T (grad T )2 ζ (div v)2 dV + k dV + T T2
(4. On the left. resulting in the conservation of entropy. using Eq. each integral term on the right must be positive.16)
This is a general equation of heat transfer (see (9. In fact.4. we can rewrite as vi ∂t vi = ∂t vi vk ∂k vi = vk ∂k 1 2 v 2
1 2 2v (v)
(4.18)
2 where v 2 = vi .3. Multiplying vi to the Navier–Stokes equation (4. Sec. A deeper insight into the entropy equation will be gained if we consider the rate of change of total entropy ρs dV in a volume V0 . the expressions on the right denote the heat production due to viscous dissipation of energy and thermal conduction. Therefore. Hence it follows that the viscosity coeﬃcients µ and ζ must be positive as well as the thermal conduction coeﬃcient k. the energy equation becomes simpler and clearer. 4. 49]: ρT (∂t s + (v · ∇)s) = σik ∂k vi + div(k grad T ).36) for the equation of thermal conduction).16) and the continuity equation. there was no term on the right-hand side.25) of an ideal ﬂuid.

the state of ﬂow is characterized by a single dimensionless number. deﬁned by Re = ρ0 UL UL = . and U/τ = U 2 /L for acceleration.52
Viscous ﬂuids
When a ﬂow of an incompressible viscous ﬂuid of density ρ0 has a single representative velocity U and a single representative length L. Denoting the dimensionless variables by primes. L. U for the dimension of velocity.8) (without the external force f ) and rewriting it with primed variables.25)
which is called the Reynolds number. if the values of Re are diﬀerent. U p = p − p0 . Equation (4. This is formulated as follows. with the reference pressure denoted by p0 . even though the sets of values U.26)
(∇ ·v = 0). the corresponding ﬂow ﬁelds are diﬀerent. τ v = v . the ﬂow ﬁelds represented by the two solutions are equivalent. L t = t . τ = L/U for time. The representative value of pressure variation is of the order of [(ρ0 L3 )(U 2 /L)/L2 ] = [ρ0 U 2 ]. z). The dimension of pressure is the same as that of [pressure] = [force]/[area] = ([mass] · [acceleration])/[area].26) thus derived is a dimensionless equation including a single dimensionless constant Re (Reynolds number). Stating it in another way. we obtain 1 ∂ v + (v · ∇ )v = −∇ p + (∇ )2 v ∂t Re (4. even though the boundary
. ν are diﬀerent. and Re is termed the similarity parameter. y. If the values of Re are the same between two ﬂows under the same boundary condition. Suppose that the space coordinates x = (x. ρ0 U 2
where the normalization is done using L as the dimension of length. This is called the Reynolds similarity law. we deﬁne x = x . Substituting these into (4. velocity v and pressure p are normalized to dimensionless variables. ν µ (4. time t.

On the contrary. the viscous term |ν∇2 v| will be larger than the inertia term |(v · ∇)v|. the magnitude of the second term of (4. If the ﬂuid viscosity is high and its kinematic viscosity ν is large enough. because a ﬂow of very small velocity U makes Re < 1 regardless of the magnitude of viscosity. 1 for ﬂows of large U .4. Taking the ratio of the two terms. large L. the ﬂow varies its state. resulting in Re < 1. The motion of a microscopic particle becomes inevitably a ﬂow of low Reynolds number because the length L is suﬃciently small. From this point of view. Such a ﬂow of low Reynolds number is called a viscous ﬂow. = 2 v|) 2 O(|ν∇ νU/L ν which is nothing but the Reynolds number. However. it will be found in the next section that the role of viscosity is still important in the ﬂows of high Reynolds numbers too. Note that the dimension of ν is the same as that of UL = [L]2 [T ]−1 with [T ] the dimension of time. Smooth ﬂows observed at low Reynolds numbers are said to be laminar. we obtain U 2/L UL O(|(v · ∇)v|) = = Re . Let us estimate relative magnitude of the second inertia term. 4. Consider a steady ﬂow in which all the ﬁeld variables do not depend on time and hence the ﬁrst term of (4.4. it will change
. Such a ﬂow is said to be a ﬂow of high Reynolds number.8). Re is also termed as the control parameter. In the ﬂows of high Reynolds numbers. Reynolds similarity law
53
conditions are the same.8) vanishes. Regarding the last viscous term. According to the increase in Reynolds number. Most ﬂows at high Reynolds numbers become turbulent. including the case of Re 1 (see Sec. and the ﬂow is mainly governed by the ﬂuid inertia. Estimating the magnitude of velocity gradient ∇v to be of the order of U/L. and at suﬃciently large values of Re . The ﬂow of Re < 1 is said to be a slow motion. Consider a sequence of states as the value of Reynolds number Re is increased gradually from a low value at the state of a laminar ﬂow.8) is |(v · ∇)v| = O(U (U/L)) = O(U 2 /L). analogous estimation leads to |ν∇2 v| = O(νU/L2 ). we will have Re or small ν. the magnitude of inertia term |(v · ∇)v| is much larger than the viscous term |ν∇2 v|.

Boundary layer
As the Reynolds number Re is increased. As an example.2. however large the value of Re . the change of velocity occurs in a thin layer of thickness δ (say) adjacent to the wall. The ﬂow can be described in the two-dimensional (x. and that the ﬂow is steady (Fig. 4. 4. The boundary
y
U(A)
U(B)
O A l B x
Fig. and a boundary layer is formed. But.6 Suppose that there is a ﬂow of an incompressible viscous ﬂuid over a plane wall AB and the ﬂow tends to a uniform ﬂow of velocity U far from the wall.
4. often written as Rc . the viscosity eﬀect can never disappear. which is termed as the critical Reynolds number. the viscous term takes small values over most part of the space of ﬂow ﬁeld. and the viscous action tends to be localized in space.e.5. we consider a plane boundary layer ﬂow. y) space. This transition to turbulence occurs at a ﬁxed value of Reynolds number. This fact means that the ﬂow in the limit of vanishing viscosity ν (i. Velocity distributions with respect to the coordinate y are schematically represented in the ﬁgure.54
Viscous ﬂuids
over to a turbulent state. In this situation.
6
Plane in ﬂuid mechanics means “two-dimensional”. ν → 0) does not necessarily coincide with the ﬂow of inviscid ﬂow (ν = 0).
Plane boundary layer.2). We take the x axis in the direction of ﬂow along the wall AB and the y axis perpendicular to AB. The velocity becomes zero on the wall y = 0 by the no-slip condition.
.

v) with the kinematic viscosity ν and the uniform density ρ. 0) as y/δ → ∞. y) space reduces to ut + uux + vuy = −(1/ρ) px + ν(uxx + uyy ). (4.31) balance each other.33)
These are consistent with the estimate of the order of magnitude just below. Equating the above two estimates in the order
. the term νuxx may be much smaller than the term νuyy in Eq.33). (4. The viscous term on the right-hand side of Eq. see Problem 4. It is useful to recognize that the scales of variation are diﬀerent in the two directions x and y. On the basis of the above estimates and the estimates below (4. Hence. and hence the term νuxx may be neglected. ux + vy = 0. (4. Within the boundary layer. (4. and the magnitude of u is given by U . the steady ﬂow in a boundary layer can be well described asymptotically by the following system of equations in the limit of small viscosity ν (Prandtl (1904).30) vanish in the steady problem under consideration. 0) (u.4 for Blasius ﬂow and Problem 4.32) (4. there is a change in the x direction however slight.4.30) (4. The two terms ux and vy in the continuity equation (4.27) (4. the Navier–Stokes equation (4. But. ux + vy = 0.29).32) can be estimated as the order O(νU/δ2 ). respectively where l δ. (4.29) and (4. Suppose that the representative scales in the directions x and y are denoted by l and δ.29) (4.31)
where the ﬁrst time derivative terms of (4.8) in the two-dimensional (x. Boundary layer
55
conditions are summarized as follows: (u. vt + uvx + vvy = −(1/ρ) py + ν(vxx + vyy ).28)
Provided that the velocity ﬁeld is expressed as (u. Both the ﬁrst and second terms on the left-hand side are estimated as the order O(U 2 /l). the change in the y direction is more rapid compared with the change in the x direction.5): uux + vuy = −(1/ρ) px + νuyy . v) = (0. v) → (U.5. at y = 0.

34)
where Re = U l/ν. the x. Parallel shear ﬂows
Consider a simple class of ﬂows of a viscous ﬂuid having only x component u of velocity v: v = (u(y. In fact. t). the convection term vanishes identically since (v · ∇)u = u∂x u = 0. however small the viscosity ν is. the pressure should be a function of x and t only: p = p(x. z. This implies that the wall contributes to the generation of vorticity. The boundary layer does not disappear. a parabolic growth of the thickness δ along the wall. (4. 0). Far from the wall.35)
The continuity equation reduces to the simple form ∂x u = 0. The left-hand side of (4. It is seen that the thickness δ of the boundary layer becomes smaller.37)
∂y p = 0. where vx is very small. This is seen from the fact that ω = vx − uy ≈ −uy (nonzero). y. or unidirectional ﬂow. l δ or δ = l ν 1 =√ . z.35).6.56
Viscous ﬂuids
of magnitude. This is called the parallel shear ﬂow.36) (4. An important diﬀerence of the ﬂow at high Reynolds numbers from the ﬂow of an inviscid ﬂuid is the existence of such a rotational layer at the boundary. z components of the Navier– Stokes equation (4. in the boundary layer there exists nonzero vorticity ω. the ﬂow velocity tends to a uniform value and therefore ω → 0 as y → ∞.8) reduce to
2 2 ∂t u − ν(∂y + ∂z )u = −
1 ∂x p ρ0
(4.
∂z p = 0. while
. More importantly.36) depends on y. In this type of ﬂows. one ﬁnds the behavior δ ∝ l. t). t. From √ the above expression. and consistent with (4.
From the last equations. 0. as the Reynolds number Re increases. we have νU U2 = 2. this is an essential diﬀerence from the inviscid ﬂow. stating that u is independent of x. Ul Re (4. Without the external force f . 4.

A solution u = u(y.36) reduces to
2 2 ∂y u + ∂z u = −
P . Parallel shear ﬂows
57
the right-hand side depends on x.1.1 at the end of this chapter. The second is the Poiseuille ﬂow [Fig. z) satisfying (4. called the Hagen–Poiseuille ﬂow which is considered as Problem 4.22)). 2µ
(4. The acceleration of a ﬂuid particle vanishes identically in steady unidirectional ﬂows. Therefore. We have another axisymmetric solution.39) is in fact an exact solution of the incompressible Navier–Stokes equation. 4.6. Writing it as P (t)/ρ0 . 4.40)
P 2 (b − y 2 ).39). 2b P = 0. Steady ﬂows
In steady ﬂows. 0). (4. these satisfy Eq.3(b)] between two parallel walls at y = ±b under a constant pressure gradient P (d = 2b in (2. we have ∂t u = 0 and P = const. The ﬁrst Couette ﬂow represents a ﬂow between the ﬁxed plate at y = −b and the plate at y = b moving with velocity U in the x direction when there is no pressure gradient [Fig. The maximum velocity U is attained at the center y = 0.41)
Obviously. hence the density does not make its appearance explicitly. 4.
. P = 0. (|y| < b) : 2D Poiseuille ﬂow. t. (|y| < b) : Couette ﬂow.6. we have − grad p = (P.38)
in which the ﬂuid is driven to the positive x direction when P is positive. 0. the equality states that both sides should be a function of t only for the consistency of the equation. Some of such solutions are as follows : uC (y) = uP (y) = U (y + b).4. The above equation (4. µ
(4.39)
where µ is the dynamic viscosity. (4.3(a)]. (4.

t and ν of the present problem. together with the boundary condition: u → ±U as y → ±∞.44): ∂t u = 2 ν∂y u.4). The velocity proﬁle at a time t can be t/t1 . 0) = U. t) depending on a similarity variable composed of three parameters y. it was a vortex sheet (see Problem 5. So that. 4. (4. Hence. From these. for y > 0 .
the wall y = 0 (Fig. 4.
Rayleigh ﬂow. one √ can form only one dimensionless variable. The x-velocity u(y. t) = 0. y/ νt. there is no characteristic length. 4.60
5
Viscous ﬂuids
4 y 3
2 4t1 1 t1
16t1
0
u 1 U
Fig.
. t) is governed by Eq.7 since the dimension of ν is [L]2 [T ]−1 . for y < 0 [Fig. (b) Diﬀusive spreading of a shear layer Let us consider a time-evolving transition layer of velocity which was initially a zero thickness plane surface.11) initially. The symmetry implies u(0. coinciding with the plane y = 0. obtained by rescaling the vertical coordinate by the ratio once the proﬁle at t1 is given. we seek a solution u(y. In this problem.
7
The same reasoning can be applied to the previous Rayleigh problem.5(a)]. −U. Suppose that u(y.4.